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Fractal behavior of tensor powers of the two dimensional space in prime characteristic

Kevin Coulembier, Pavel Etingof, Victor Ostrik and Daniel Tubbenhauer K.C.: The University of Sydney, School of Mathematics and Statistics F07, Office Carslaw 717, NSW 2006, Australia, www.maths.usyd.edu.au/u/kevinc, ORCID 0000-0003-0996-3965 [email protected] P.E.: Department of Mathematics, Room 2-282, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA 02139, USA, math.mit.edu/ etingof/, ORCID 0000-0002-0710-1416 [email protected] V.O.: University of Oregon, Department of Mathematics, Eugene, OR 97403, USA,
pages.uoregon.edu/vostrik, ORCID 0009-0002-2264-669X [email protected] D.T.: The University of Sydney, School of Mathematics and Statistics F07, Office Carslaw 827, NSW 2006, Australia, www.dtubbenhauer.com, ORCID 0000-0001-7265-5047 [email protected]

Abstract.

We study the number of indecomposable summands in tensor powers of the vector representation of SL2. Our main focus is on positive characteristic where this sequence of numbers and its generating function show fractal behavior akin to Mahler functions.

Key words and phrases:

Tensor products, asymptotic behavior, fractals, Mahler functions, subcategories of finite tensor categories.

2020 Mathematics Subject Classification:

Primary: 11N45, 18M05; Secondary: 11B85, 26A12, 30B30.

1. Introduction

Let $\mathbf{k}$ be a field and let $\Gamma$ be a group, possibly an algebraic group or affine group scheme over $\mathbf{k}$ . For any finite dimensional $\Gamma$ -representation $W$ over $\mathbf{k}$ let $\nu(W)\in\mathbb{Z}_{\geq 0}$ be an integer such that

\displaystyle W\cong{\textstyle\bigoplus_{i=1}^{\nu(W)}}W_{i},

where $W_{i}$ are indecomposable $\Gamma$ -representations. This number is well-defined by the Krull–Schmidt theorem. Now let $V$ be a finite dimensional $\Gamma$ -representation and define the integer sequence

\displaystyle b_{n}=b_{n}(V):=\nu(V^{\otimes n}),\;n\in\mathbb{Z}_{\geq 0}.

The study of the growth of $b_{n}$ is the main motivation of this paper.

The related question to study the growth of $\ell(V^{\otimes n})$ , where $\ell(W)$ is the length of $W$ , is usually much easier and we discuss this along the way, e.g. in Section 2A and Section 2BV below.

We write $\sim$ for ‘asymptotically equal’. One could hope that

(1.1)

\displaystyle b_{n}\sim h(n)\cdot n^{\tau}\cdot\beta^{n},\quad\begin{aligned} &h\colon\mathbb{Z}_{\geq 0}\to\mathbb{R}_{>0}\text{ is a function \emph{bounded away from $0,\infty$}},\\[-2.84544pt] &n^{\tau}\text{ is the \emph{subexponential factor}, $\tau\in\mathbb{R}$},\\[-2.84544pt] &\beta^{n}\text{ is the \emph{exponential factor}, $\beta\in\mathbb{R}_{\geq 1}$}.\end{aligned}

In practice, $h(n)$ is often a constant or alternates between finitely many constants, but sometimes $h(n)$ is more complicated.

We do not know in what generality Equation 1.1 holds, but expressions of this form are very common in the theory of asymptotics of generating functions, see for example [FS09] or [Mis20] for the relation between counting problems and the analysis of generating functions.

1A. The exponential factor

Even without assuming (1.1) one can define the value $\beta:=\lim_{n\to\infty}\sqrt[n]{b_{n}(V)}$ , and it is proven in [COT24, Theorem 1.4] that

(1A.1)

\displaystyle\beta=\dim_{\mathbf{k}}V.

Thus, Equation 1A.1 gives $(\dim_{\mathbf{k}}V)^{n}$ as the exponential factor of the growth rate of $b_{n}$ .

The goal of this paper is to provide a more precise asymptotics, that is, one of the form Equation 1.1, for the sequence $b_{n}$ in the case $\dim_{\mathbf{k}}V=2$ and $\Gamma=GL(V)$ or, equivalently, $\Gamma=SL(V)$ . This is a nontrivial case where one can hope to understand the asymptotics explicitly. In this setting the subexponential factor is determined by Section 1B, as well as Section 2A and Section 2A.

We do not calculate $h(n)$ , but it appears to be oscillating, cf. Section 1B. See also Section 8 where we determine $h(n)$ for $p=2$ .

1B. The main theorem – the subexponential factor

If $\mathbf{k}$ is finite, then $\Gamma=SL_{2}(\mathbf{k})$ is a finite group and the asymptotic behavior of $b_{n}=b_{n}(\mathbf{k}^{2})$ can be easily obtained, see for example Section 2A below or [LTV23, Example 8] which settle the case of a finite field. We give an example which is prototypical in this situation:

Example \theExample.

For $\mathbf{k}=\mathbb{F}_{3}$ we get

\displaystyle(b_{n})_{n\in\mathbb{Z}_{\geq 0}}=(1,1,2,2,6,6,22,22,86,\dots)\quad\text{and}\quad b_{n}\sim\left(\frac{1}{6}+\frac{(-1)^{n}}{12}\right)\cdot n^{0}\cdot 2^{n}.

(We highlight that the subexponential factor is $n^{0}=1$ .) Note that $\lim_{n\to\infty}\frac{b_{n}}{2^{n}}$ does not exist, but the ratio $\frac{b_{n}}{2^{n}}$ is rather governed by the periodic function $h(n)=\frac{1}{6}+\frac{(-1)^{n}}{12}$ .

From now on let $V$ be a two dimensional vector space over an infinite field. Let $\Gamma$ be the algebraic group $SL(V)\simeq SL_{2}(\mathbf{k})$ . Let $b_{n}=b_{n}(V)$ be the sequence as above. As a matter of fact, the sequence $b_{n}$ depends only on the characteristic of $\mathbf{k}$ (which is only implicit in our notation), so we only need to study it for one infinite field for each $p\geq 0$ . That is, we have

(1B.1)

\displaystyle b_{n}=\text{number of indecomposable $SL_{2}(\mathbf{k})$-representations in $(\mathbf{k}^{2})^{\otimes n}$},

where $\mathbf{k}$ is either $\mathbb{C}$ for $p=0$ or $\bar{\mathbb{F}}_{p}$ otherwise.

The following example is classical, and settles the case $p=0$ :

Example \theExample.

Assume $p=0$ . Then $b_{n}$ is the sequence of middle binomial coefficients

\displaystyle(b_{n})_{n\in\mathbb{Z}_{\geq 0}}=\left(\binom{n}{\lfloor n/2\rfloor}\right)_{n\in\mathbb{Z}_{\geq 0}}=(1,1,2,3,6,10,20,35,70,\dots)

Then Stirling’s formula implies

\displaystyle b_{n}\sim\sqrt{2/\pi}\cdot n^{-1/2}\cdot 2^{n},

as one easily checks. In this case the periodic function is constant, i.e. $h(n)=\sqrt{2/\pi}=\lim_{n\to\infty}\frac{b_{n}}{n^{-1/2}\cdot 2^{n}}$ . Since this situation is semisimple, the same formula works for the length instead of the number of indecomposable summands.

For the remainder of the paper let $p>0$ . Then the asymptotic behavior of the sequence $b_{n}$ turns out to be more complicated. Namely let us define

(1B.2)

\displaystyle t_{p}:=-\frac{1}{2}\log_{p}\frac{2p^{2}}{p+1}=\frac{1}{2}\frac{\ln(p+1)-\ln(2p^{2})}{\ln p}=\frac{1}{2}\left(\log_{p}\frac{p+1}{2}\right)-1.

As we will see, $t_{p}$ plays the role of $\tau$ in Equation 1.1. To give some numerical expressions, we have for example:

	$\displaystyle t_{2}=-0.7075187496\ldots\,,\quad t_{3}=-0.6845351232\ldots\,,\quad t_{5}=-0.6586969028\ldots\,,$
	$\displaystyle t_{7}=-0.6437928129\ldots\,,\quad t_{101}=-0.5740278484\ldots\,.$

It is easy to see that all $t_{p}$ are irrational. Moreover, all these numbers are transcendental, as follows from the (logarithm form of the) Gelfond–Schneider theorem which implies that $\log_{p}r$ , for $p$ a prime number and $r\in\mathbb{Q}$ , is transcendental unless $r$ is a power of $p$ .

Remark \theRemark.

By very similar arguments, all $\tau$ that we will see below that are not manifestly rational will be irrational and even transcendental.

Notation \theNotation.

Throughout, we will use big O notation (or rather its generalization often called Bachmann–Landau notation). Most important for us from this family of notations are $\sim$ as above, small $o$ , big $O$ , big $\Theta$ and $\asymp$ . That is, given two real-valued functions $f$ , $g$ which are both defined on $\mathbb{Z}_{\geq 0}$ or on $D\subset\mathbb{R}$ for the final point, we will write

(1B.3)

\displaystyle\begin{aligned} f\sim g&\;\Leftrightarrow\;\forall\varepsilon>0,\,\exists n_{0}\;\text{ such that }\;\framebox{$|\tfrac{f(n)}{g(n)}-1|<\varepsilon$},\;\forall n>n_{0},\\ f\in o(g)&\;\Leftrightarrow\;\forall C>0,\,\exists n_{0}\;\text{ such that }\;\framebox{$|f(n)|\leq C\cdot g(n)$},\;\forall n>n_{0},\\ f\in O(g)&\;\Leftrightarrow\;\exists C>0,\,\exists n_{0}\;\text{ such that }\;\framebox{$|f(n)|\leq C\cdot g(n)$},\;\forall n>n_{0},\\ f\in\Theta(g)&\;\Leftrightarrow\;\exists C_{1},C_{2}>0,\,\exists n_{0}\;\text{ such that }\;\framebox{$C_{1}\cdot g(n)\leq f(n)\leq C_{2}\cdot g(n)$},\;\forall n>n_{0},\\ f\asymp_{D}g&\;\Leftrightarrow\;\exists C>1\;\text{ such that }\;\framebox{$C^{-1}\cdot g(x)\leq f(x)\leq C\cdot g(x)$},\;\forall x\in D.\end{aligned}

We sometimes omit the subscript $D$ if no confusion can arise.

Our main result is:

Main Theorem \theMain.Theorem.

For any $p>0$ there exist $C_{1}=C_{1}(p),C_{2}=C_{2}(p)\in\mathbb{R}_{>0}$ such that we have

\displaystyle C_{1}\cdot n^{t_{p}}\cdot 2^{n}\leq b_{n}\leq C_{2}\cdot n^{t_{p}}\cdot 2^{n},\qquad n\geq 1.

Thus, $(n\mapsto b_{n})\in\Theta(n^{t_{p}}\cdot 2^{n})$ .

Moreover, for $p=2$ we have a quite complete picture and we prove stronger results: we show that Equation 1.1 holds and we determine $h(n)$ , see Section 7 and Section 8. We expect that similar methods can be used to handle the case $p>2$ .

Remark \theRemark.

As usual in modular representation theory, the case $p=0$ behaves like $p=\infty$ . Indeed, we have

\displaystyle t_{\infty}:=\lim_{p\to\infty}t_{p}=-\frac{1}{2},

so we can compare Section 1B with Section 1B.

Remark \theRemark.

In contrast to characteristic zero as in Section 1B, the limit

\displaystyle\lim_{n\to\infty}\frac{b_{n}}{n^{t_{p}}\cdot 2^{n}}

does not exist, i.e. for $p>0$ there is no constant $c$ such that $b_{n}\sim c\cdot n^{t_{p}}\cdot 2^{n}$ .

Example \theExample.

For $p=2$ we have $b_{2n-1}=b_{2n}$ and the logplot of the even values of $b_{n}/(n^{-0.708}\cdot 2^{n})$ (splitting the sequence $b_{n}/(n^{-0.708}\cdot 2^{n})$ into even and odd makes it more regular, see for example Section 5) gives

\displaystyle\leavevmode\hbox to229.98pt{\vbox to149.04pt{\pgfpicture\makeatletter\hbox{\hskip 114.99016pt\lower-74.51895pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-111.65715pt}{-71.18594pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=142.26378pt]{figs/plottt2-2}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-71.2956pt}{46.21683pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{even $n$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\leavevmode\hbox to232.87pt{\vbox to149.04pt{\pgfpicture\makeatletter\hbox{\hskip 116.43555pt\lower-74.51895pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-113.10254pt}{-71.18594pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=142.26378pt]{figs/plottt2-3}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-84.49007pt}{46.21683pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{even $n$ zoom}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.

These illustrations show the graph of $b_{n}/(n^{-0.708}\cdot 2^{n})$ for $n\in\{0,2,\dots,198,200\}$ and $n\in\{0,2,\dots,1998,2000\}$ (the fluctuations in the beginning vanish quickly) and we see the tiny oscillation, which is proven to be true in Section 8.

Remark \theRemark.

For $p=2$ Section 1B, or rather the stronger version in Section 7B, was independently proven in [La24] using quite different methods. Moreover, and similar but different to Section 8, the paper [La24] also identifies $h(n)$ from Equation 1.1. For $p=2$ [La24] even implies some results for other representations as well.

Remark \theRemark.

Let us comment on the quantum case; the analog of Section 1B for quantum $SL_{2}$ . If the quantum parameter is not a root of unity, then the same discussion as in Section 1B works. The next case, where the quantum parameter is a root of unity and the underlying field is of characteristic zero can be deduced from [LPRS24]. For the final possibility, the so-called mixed case as e.g. in [STWZ23], we expect a result very similar to Section 1B.

1C. Proof outline of Section 1B

It is a classical fact that all direct summands of $V^{\otimes n}$ are tilting representations over $\Gamma=SL(V)$ . In the special case of $\Gamma=SL_{2}(\mathbf{k})$ the characters of indecomposable tilting representations are explicitly known; using this we derive a recursion for the sequence $b_{n}$ , see Section 3B.

Then we translate this recursion into a functional equation for the generating function $F(z)$ of the sequence $b_{n}$ , see Section 3A. Then we analyze the behavior of the function $F(z)$ in the vicinity of its radius of convergence and deduce Section 1B by employing suitable Tauberian theorems in Section 6.

One of the steps is the following elementary looking inequality from Section 5A:

\displaystyle b_{n+2}\leq 4b_{n},\;n\geq 0.

1D. Odds and ends

Here are a few open questions that might be interesting to explore. (i) is proven for $p=2$ , see Section 7 and Section 8, and similar methods apply in general.

(i)

In Section 5A we prove that $b_{n+2}\leq 4b_{n}$ . A natural question is whether we have $\lim_{n\to\infty}b_{n+2}/b_{n}=4$ . Moreover, recall from Section 1B that there is no constant such that $b_{n}\sim c\cdot n^{t_{p}}\cdot 2^{n}$ . However, one could conjecture that there exists a continuous real function $\tilde{h}(x)$ of period one such that

(1D.1) $\displaystyle b_{n}\sim\tilde{h}\big{(}\log_{2p}(n)\big{)}\cdot n^{t_{p}}\cdot 2^{n}.$

For $p=2$ this is proven in Section 7B.
(ii)

Recall from Section 1B that $\frac{b_{n}}{n^{t_{p}}\cdot 2^{n}}$ is bounded by two constants, and it would be interesting to have good explicit values for these constants, see Section 9D for some possible values. For a more precise asymptotic formula one would need to analyze the oscillation as in Section 1B.
(iii)

One could try to compute other asymptotic formulas. An example that comes to mind is to verify an analog of Section 1B (with the same subexponential factor) for the three dimensional tilting $SL_{2}$ -representation $\mathrm{Sym}^{2}V$ (in case $p\neq 2$ ). When $p=2$ this three dimensional $SL_{2}$ -representation is not tilting and getting asymptotic formulas might be very hard.

Acknowledgments. We thank Michael Larsen for sharing a draft of [La24], which uses very interesting methods that are quite different from the ones in this work. The methods are exciting and important, so we all agreed that it makes sense to write two separate papers. We also thank Henning Haahr Andersen, David He, Abel Lacabanne and Pedro Vaz for useful discussions and comments, and are very grateful to Kenichi Shimizu for pointing out a gap in the proof of Appendix A in the first version.

We express our appreciation to ChatGPT for their assistance with proofreading. In addition, D.T. extends heartfelt thanks to the tree constant for inspiration.

P.E.’s work was partially supported by the NSF grant DMS-2001318, D.T. was supported by the ARC Future Fellowship FT230100489.

2. Fractal behavior of growth problems

Before coming to the main parts of this paper, let us briefly indicate the main difficulty (and maybe the most exciting part) of growth problems in the above sense: a certain type of fractal behavior of these sequences and of their generating functions. For the sake of this paper, and partially justified by the discussion in this section, we use fractal behavior to mean that the exponent $\tau$ of the subexponential factor is transcendental (note that e.g. $t_{p}$ from Equation 1B.2 is transcendental), or at least irrational. For instance, $t_{p}$ could be the fractal dimension of some fractal.

2A. No fractal behavior

For $\mathbf{k}$ of characteristic zero, let $\Gamma$ be a connected reductive algebraic group over $\mathbf{k}$ . [Bia93, Theorem 2.2] and [CEO24, Theorem 2.5] give the asymptotic for the numbers $b_{n}$ for an arbitrary finite dimensional $\Gamma$ -representation. The example of $\Gamma=SL_{2}(\mathbf{k})$ and its vector representation is given in Section 1B where $b_{n}\sim\sqrt{2/\pi}\cdot n^{-1/2}\cdot 2^{n}$ .

In general, [Bia93, Theorem 2.2] and [CEO24, Theorem 2.5] prove (since this case is semisimple the same holds true for the length instead of the number of indecomposable summands):

Proposition \theProposition.

In the above setting, the asymptotic takes the form

\displaystyle b_{n}\sim C\cdot n^{\tau}\cdot(\dim_{\mathbf{k}}V)^{n}\text{ with }\tau\in\tfrac{1}{2}\mathbb{Z},C\in\mathbb{R}_{>0}.

Thus, the subexponential factor $n^{\tau}$ always has some half integer exponent.∎

It is not a coincidence that $\tau\in\tfrac{1}{2}\mathbb{Z}$ since:

“The nature of the generating function’s singularities determines the associated subexponential factor.”

The above strategy is well-known in symbolic dynamics, and under certain assumptions on the singularities of the generating function one always gets half integer powers for the subexponential growth term. For example, this works if the generating function is algebraic.

In fact, there are well developed algorithms to compute the asymptotics if the generating function is sufficiently nice (e.g. meromorphic), see for example [Mis20, Section 7.7]. The algorithm presented therein always produces an exponent $s\in\tfrac{1}{2}\mathbb{Z}$ and is enough to treat the case of a connected reductive algebraic group in characteristic zero.

Here is another example where one always gets a half integer coefficient, namely zero, regardless of the characteristic. As we will argue below, this means that these cases do not show fractal behavior with respect to the growth problems we consider.

Let $\Gamma$ be a finite group and let $V$ be a representation of $\Gamma$ over $\mathbf{k}$ , with $\mathbf{k}$ of arbitrary characteristic. We assume for simplicity that $V$ is a faithful $\Gamma$ -representation; if $V$ is not faithful we replace $\Gamma$ by a quotient and continue as below. Let $M\subset\Gamma$ be the central cyclic subgroup of $\Gamma$ containing all scalar matrices. Let $m=|M|$ . The category $\mathrm{Rep}(\Gamma)$ of finite dimensional $\Gamma$ -representations splits into a direct sum

\displaystyle\mathrm{Rep}(\Gamma)=\bigoplus_{i\in\mathbb{Z}/m\mathbb{Z}}\mathrm{Rep}(\Gamma)_{i},

where $M$ acts on objects of $\mathrm{Rep}(\Gamma)_{i}$ in the same way as in $V^{\otimes i}$ .

Let $\mathrm{S}^{i}(\Gamma)$ be the set of isomorphism classes of simple objects of $\mathrm{Rep}(\Gamma)_{i}$ ; for any $L\in\mathrm{S}^{i}(\Gamma)$ let $P(L)\in\mathrm{Rep}(\Gamma)_{i}$ be its projective cover. For any $i\in\mathbb{Z}/m\mathbb{Z}$ we define

\displaystyle\ell_{i}=\frac{m}{|\Gamma|}\bigg{(}\sum_{L\in\mathrm{S}^{i}(\Gamma)}\dim_{\mathbf{k}}P(L)\bigg{)},\quad\nu_{i}=\frac{m}{|\Gamma|}\bigg{(}\sum_{L\in\mathrm{S}^{i}(\Gamma)}\dim_{\mathbf{k}}L\bigg{)}.

Let $A$ be the regular representation of $\Gamma$ and let $A=\oplus_{i\in\mathbb{Z}/m\mathbb{Z}}A_{i}$ be its decomposition into summands $A_{i}\in\mathrm{Rep}(\Gamma)_{i}$ . A result of Bryant–Kovács [BK72, Theorem 2] says that for sufficiently large $n$ with $n\equiv i\bmod{m}$ we have that $A_{i}$ is a direct summand of $V^{\otimes n}$ . Similarly to $b_{n}$ , let $l_{n}=l_{n}(V):=\ell(V^{\otimes n})$ , where $\ell({}_{-})$ denotes the length of representations. Assuming $\mathbf{k}=\overline{\mathbf{k}}$ , it follows easily that for $n\equiv i\bmod{m}$ we have

\displaystyle l_{n}\sim\ell_{i}\cdot(\dim_{\mathbf{k}}V)^{n},\quad b_{n}\sim\nu_{i}\cdot(\dim_{\mathbf{k}}V)^{n}.

In particular, each of the sequences $l_{n}/(\dim_{\mathbf{k}}V)^{n}$ and $b_{n}/(\dim_{\mathbf{k}}V)^{n}$ has at most $m$ limit points. In particular, [Mis20, Section 7.7] applies and the subexponential factor has half integer exponent. A similar analysis works without the assumption $\mathbf{k}=\overline{\mathbf{k}}$ .

This immediately proves that $h(n)$ , in this case, is periodic with period $m$ , the subexponential factor is trivial and the exponential factor is given by the dimension. Precisely:

Proposition \theProposition.

For a finite group $\Gamma$ and a $\Gamma$ -representation $V$ we always have

\displaystyle\begin{aligned} l_{nm+r}&\sim s(r)\cdot n^{0}\cdot(\dim_{\mathbf{k}}V)^{nm},\\ b_{nm+r}&\sim t(r)\cdot n^{0}\cdot(\dim_{\mathbf{k}}V)^{nm},\end{aligned}\quad\text{with }s(r),t(r)\in(0,1],t(r)\leq s(r),

for some $m\in\mathbb{Z}_{\geq 0}$ and all $r\in\{0,\dots,m-1\}$ .∎

The scalars $s(r),t(r)$ in Section 2A are easy to compute. For $t(r)$ see for example [CEO24, Proposition 2.1] and [LTV23, (2A.1)] for characteristic zero, and [LTV24, Section 5] for arbitrary characteristic.

Example \theExample.

For instance if $p=2$ , and $\Gamma$ is the symmetric group $S_{3}$ or $S_{4}$ and $V$ is a faithful representation of $\Gamma$ we have $l_{n}\sim\frac{2}{3}\cdot\dim_{\mathbf{k}}V^{n}$ and $b_{n}\sim\frac{1}{2}\cdot\dim_{\mathbf{k}}V^{n}$ .

Remark \theRemark.

Let $\mathbf{C}$ be a finite tensor category, see e.g. [EGNO15, Chapter 6] for details. Let $X$ be a tensor generator of $\mathbf{C}$ . We have a decomposition of $\mathbf{C}$ over the universal grading group $U$ of $\mathbf{C}$ . Write $X\cong\oplus_{i\in I}X_{i}$ for $X_{i}$ indecomposables with $X_{i}$ of degree $d_{i}\in U$ . Let $U_{X}$ be the subgroup of $U$ generated by $d_{i}-d_{j}$ . Then $U/U_{X}$ is a cyclic group $\mathbb{Z}/m\mathbb{Z}$ , and statements similar to Section 2A hold. Details are omitted, but the key statement hereby is proven in Appendix A in the appendix.

To summarize, in the two above settings the subexponential exponent $\tau$ is a half-integer, in particular not transcendental, and the function $h(n)$ is constant up to some period $m\in\mathbb{Z}_{\geq 0}$ .

2B. Fractal behavior

For $\Gamma=SL_{2}(\mathbf{k}^{2})$ our problem, where $\mathbf{k}$ is of prime characteristic, is difficult also because the generating function that we compute in Section 3A does not have nice enough properties to run the classical strategies. For example, we will see that we have to face a dense set of singularities, see e.g. Section 4B. And in fact, the exponent we get is not a half integer but rather the transcendental number $t_{p}$ from Equation 1B.2.

In a bit more details, see Section 3C for the precise statement, we get a functional equation for the generating function $F$ of the numbers $b_{n}$ that takes the form

(2B.1)

\displaystyle F(w)=r_{1}(w)+r_{2}(w)\cdot F(w^{p}).

Functions of this type are called 2-Mahler functions of degree $p$ .

Remark \theRemark.

The name originates in Mahler’s approach to transcendence and algebraic independence results for the values at algebraic points in the study of power series satisfying functional equations of a certain type. Mahler’s original functional equation is of the form $F(w)=w+F(w^{2})$ , which is an example of what we call a 2-Mahler function of degree 2.

Let $r_{i}(w)$ denote rational functions. The Mahler functions above have been generalized under the umbrella of $s$ -Mahler functions of degree $p$ satisfying

(2B.2)

\displaystyle r_{0}(w)\cdot F(w)=r_{1}(w)+r_{2}(w)\cdot F(w^{p})+r_{3}(w)\cdot F(w^{p^{2}})+\dots+r_{s}(w)\cdot F(w^{p^{s-1}}),

but we will not need this generalization. We refer to [Nis96] for a nice discussion of Mahler functions and a list of historical references. Mahler functions occur in combinatorics as generating functions of partitions and related structures.

A crucial fact is that such a Mahler function often grows with exponent $\tau+1$ for a transcendental $\tau$ when approaching its relevant singularity. We will use this in Section 4.

Moreover, connections from Mahler functions that grow with transcendental $\tau$ to fractals are well-studied. To the best of our knowledge, general theorems relating them are not known, and connections are only example-based. In the rest of this section, we give several examples that are easier to deal with than our main result.

The common source for these examples is the well-known principle that projecting $p$ -adic objects onto the real world leads to fractals. At the same time, modular representations of algebraic groups are known to exhibit $p$ -adic patterns, stemming from Steinberg-type tensor product theorems. Therefore, one can expect that combinatorial invariants of modular representations of algebraic groups (such as dimensions, weight multiplicities, etc.) tend to exhibit fractal behavior. In fact, fractal behavior in the study of reductive groups in prime characteristic has been observed several times, and is part of folk knowledge.

Most relevant for this paper are fractal patterns within the study of tilting representations. For example, for an algebraically closed field $\mathbf{k}$ of prime characteristic $p$ , let $\Gamma=SL_{2}(\mathbf{k})$ . The multiplicities of Weyl in tilting $\Gamma$ -representations have fractal patterns of step size $p$ and so do their characters, by Donkin’s tensor product formula [Don93, Proposition 2.1] (this is a bit easier to see in the reformulation given in [STWZ23, Section 3]). The same is reflected in the Temperley–Lieb combinatorics, see e.g. [BLS19, Theorem 2.8] or [Spe23, Figure 3], which is exploited in [KST24].

For higher rank there are also several known instances of fractal behavior of tilting characters, see for example the picture of cells in [And04, Figure 1] or billiards of tilting characters as in [LW18, Section 4].

2BI. Modular representations and the Cantor set

One of the simplest examples of a fractal is the (bounded) Cantor set $\mathtt{C}$ , which can be realized as the set of real numbers in $[0,1]$ which admit a ternary expansion without digit $1$ with the probably familiar picture

\displaystyle\leavevmode\hbox to199.63pt{\vbox to63.76pt{\pgfpicture\makeatletter\hbox{\hskip 99.81346pt\lower-31.87965pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-96.48045pt}{-28.54665pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=56.9055pt]{figs/cantor-example1}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.

This example is related to representation theory of $\Gamma=SL_{2}(\mathbf{k})$ for $\mathbf{k}$ being algebraically closed of characteristic $p$ (the standard choice is $\mathbf{k}=\bar{\mathbb{F}}_{p}$ ) as follows.

The finite dimensional simple $\Gamma$ -representations $L_{n}$ are indexed by $n\in\mathbb{Z}_{\geq 0}$ , their highest weight. Expressing the number $p$ -adically, say $n=[n_{r},\dots,n_{1},n_{0}]=n_{r}p^{r}+\dots+n_{1}p+n_{0}$ with $n_{i}\in\{0,\dots,p-1\}$ , one gets

\displaystyle L_{n}\cong L_{n_{r}}^{(r)}\otimes\dots\otimes L_{n_{1}}^{(1)}\otimes L_{n_{0}},

where the exponent ⁽ⁱ⁾ denotes Frobenius twists which acts on characters by $x\mapsto x^{p^{i}}$ . This is a special case of Steinberg’s tensor product theorem. Since the character of $L_{n}$ for $n=[n_{0}]$ is $\frac{x^{n}-x^{-n}}{x-x^{-1}}$ , this gives a description of the weights of $L_{n}$ for $n\in\mathbb{Z}_{\geq 0}$ .

This also shows that we may consider, often infinite dimensional, representations $L_{n}$ where $n\in\mathbb{Z}_{p}$ is a $p$ -adic integer, which is made explicit in Haboush’s generalized Steinberg tensor product theorem for distribution algebras [Hab80, Theorem 4.9]. Concretely, given $n\in\mathbb{Z}_{p}$ , we may define $L_{n}$ to be the infinite tensor product

\displaystyle L_{n}:=\bigotimes_{j=0}^{\infty}L_{n_{j}}^{(j)},\quad\mbox{with}\quad n=[\dots,n_{r},\dots,n_{1},n_{0}]=\sum_{j=0}^{\infty}n_{j}p^{j},

which, by definition, is the span of tensor products of vectors in $L_{n_{j}}^{(j)}$ such that almost all of them are (fixed) highest weight vectors.

The group $\Gamma$ does not act in this space, but $L_{n}$ admits an action of the distribution algebra $\mathrm{Dist}=\mathrm{Dist}(\Gamma)$ , for which this representation is generated by the highest weight vector $v_{n}$ . This gives $L_{n}$ a $\mathbb{Z}_{\geq 0}$ -grading with even degrees, placing vectors of weight $\mu$ in degree $n-\mu$ . The Hilbert series for this grading is thus

\displaystyle h_{L_{n}}(w)=\prod_{j=0}^{\infty}(1+w^{2\cdot p^{j}}+\dots+w^{2n_{j}\cdot p^{j}}).

This is a holomorphic function for $|w|<1$ , and we will mostly consider it on the interval $(0,1)$ , i.e. as $h_{L_{n}}\colon(0,1)\to\mathbb{R}$ .

Consider now the special case $p=3$ , $n=-\frac{1}{2}$ . Hence, $n_{j}=1$ for all $j$ and we get

\displaystyle h(w):=h_{L_{-1/2}}(w)=\prod_{j=0}^{\infty}(1+w^{2\cdot 3^{j}}).

This function satisfies the functional equation

\displaystyle h(w)=(1+w^{2})\cdot h(w^{3}).

This is a 2-Mahler function of degree $3$ , and the relevant singularity is at $w=1$ .

The Taylor coefficients of $h(w)$ form the Cantor set sequence $(\mathrm{ca}_{n})_{n\in\mathbb{Z}_{\geq 0}}$ defined by

\displaystyle\mathrm{ca}_{n}=\begin{cases}1&\text{if the ternary expansion of $n$ contains no $1$},\\ 0&\text{otherwise}.\end{cases}

This sequence is well-studied, see e.g. [CE21, Section 1] and [OEI23, A292686], and $h(w)$ is its generating function.

The corresponding fractal is constructed as follows. Let $D(L)$ denote the set of degrees of $L$ , so that $D(L_{-1/2})$ is the set of nonnegative integers with ternary representations where all digits of $m$ take values $0,2$ only. For $N\in\mathbb{Z}_{\geq 0}$ , define $\Delta_{N}:=\frac{1}{3^{N}}D(L_{-1/2})$ . We then have $\Delta_{N}\subset\Delta_{N+1}$ . Let $\Delta_{\infty}:=\cup_{N\in\mathbb{Z}_{\geq 0}}\Delta_{N}$ . Then the closure of $\Delta_{\infty}$ in $\mathbb{R}$ is the unbounded Cantor set $\mathtt{C}_{\infty}$ , which is invariant under multiplication by $3$ . The set $\mathtt{C}$ is the intersection $\mathtt{C}_{\infty}\cap[0,1]$ , and $\mathtt{C}_{\infty}=\cup_{N\in\mathbb{Z}_{\geq 0}}3^{N}\mathtt{C}$ (nested union). Note that the set $\mathtt{C}_{\infty}$ comes with a natural measure $\mu$ , the Hausdorff measure of $\mathtt{C}_{\infty}$ : the weak limit of the counting measures of $\Delta_{N}$ rescaled by dividing by $|\Delta_{N}\cap[0,1]|=2^{N}$ .

Now, by the standard theory of Mahler functions, see e.g. [BC17, Theorem 1], we get

\displaystyle C_{1}(1-w)^{-\tau}\big{(}1+o(1)\big{)}\leq h(w)\leq C_{2}(1-w)^{-\tau}\big{(}1+o(1)\big{)},\ x\uparrow 1,

for some $0<C_{1}<C_{2}$ with $C_{1},C_{2}\in\mathbb{R}$ . Moreover, $\tau$ is found by substituting $(1-w)^{-\tau}$ into the Mahler equation, i.e. from the condition $(1-w)^{-\tau}\sim(1+w^{2})(1-w^{3})^{-\tau}$ as $w\uparrow 1$ . This yields $1=2\cdot 3^{-\tau}$ , hence

\displaystyle\tau=\log_{3}2\approx 0.631.

The number $\tau$ is transcendental, and the Hausdorff (and Minkowski) dimension of the Cantor sets $\mathtt{C}$ and $\mathtt{C}_{\infty}$ .

Rewriting the above using $\asymp$ , see Equation 1B.3 for the notation, we get

\displaystyle h(w)=\sum_{n\in\mathbb{Z}_{\geq 0}}\mathrm{ca}_{n}w^{n}\asymp_{w\uparrow 1}(1-w)^{-\tau}\big{(}1+o(1)\big{)}.

Then Tauberian theory (as for example in Section 6A) implies

\displaystyle\sum_{k=0}^{n}\mathrm{ca}_{k}\asymp n^{\tau}\big{(}1+o(1)\big{)}.

This describes the asymptotics of Cesáro sums of the Cantor set sequence, which counts the dimension of the subspace $L_{-1/2}[\leq n]$ spanned by vectors in $L_{-1/2}$ of degree $\leq n$ .

One may further ask how the sequence $C_{n}=n^{-\tau}\sum_{k=0}^{n}\mathrm{ca}_{k}$ behaves when $n\to\infty$ or, related, how the function $(1-x)^{\tau}h(w)$ (equivalently, $\ln(w^{-1})^{\tau}h(w)$ ) behaves when $w\uparrow 1$ . This behavior can be analyzed as follows. Let $\theta(m)=1$ for $m<0$ and $\theta(m)=0$ for $m\geq 0$ . We have

\displaystyle\lim_{k\to\infty}\ln(w^{-3^{-k}})^{\tau}h(w^{3^{-k}})=\ln(w^{-1})^{\tau}\lim_{k\to\infty}2^{-k}\prod_{m=-k}^{\infty}(1+w^{2\cdot 3^{m}})=h_{0}(w),

for the function

\displaystyle h_{0}(w):=\ln(w^{-1})^{\tau}\prod_{m=-\infty}^{\infty}\frac{1+w^{2\cdot 3^{m}}}{2^{\theta(m)}}.

This is a periodic function in the sense that $h_{0}(w)=h_{0}(w^{3})$ (note that the product is absolutely convergent). This implies that the function $\ln(w^{-1})^{\tau}h(w)$ has no limit as $w\uparrow 1$ and asymptotically has oscillatory behavior: it approaches the periodic function $h_{0}(w)$ .

Plotting this illustrates the overall growth rate and the oscillation:

(2B.3)

\displaystyle\leavevmode\hbox to203.24pt{\vbox to120.13pt{\pgfpicture\makeatletter\hbox{\hskip 101.62021pt\lower-60.06496pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-98.2872pt}{-56.73195pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=113.81102pt]{figs/cantor3}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.36137pt}{26.20276pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=3$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\leavevmode\hbox to229.98pt{\vbox to120.13pt{\pgfpicture\makeatletter\hbox{\hskip 114.99016pt\lower-60.06496pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-111.65715pt}{-56.73195pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=113.81102pt]{figs/cantor3b}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.36137pt}{26.20276pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=3$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.

To analyze this a bit further, it is easy to see that the analytic function $\mathtt{h}_{0}(v):=v^{-\tau}h_{0}(e^{-v})$ in the region $\mathrm{Re}\,v>0$ , which satisfies the equation $2\mathtt{h}_{0}(3v)=\mathtt{h}_{0}(v)$ , is nothing but the Laplace transform of the Hausdorff measure $\mu$ of the set $\mathtt{C}_{\infty}$ :

\displaystyle\mathtt{h}_{0}(v)=\int_{0}^{\infty}e^{-tv}d\mu(t).

This implies that

\displaystyle h_{0}(w)=\ln(w^{-1})^{\tau}\int_{0}^{\infty}w^{t}d\mu(t).

Similarly, the sequence $C_{n}$ for large $n$ behaves as the devil’s staircase function:

\displaystyle\lim_{m\to\infty}C_{\lfloor 3^{m}w\rfloor}=w^{-\tau}\mu([0,w]),

which is periodic under the map $w\mapsto 3w$ . This is visualized in Equation 2B.3. Hence, this sequence does not have a limit as $n\to\infty$ and exhibits oscillatory behavior, approaching the periodic function $\log_{3}(n)^{-\tau}\mu\big{(}[0,\log_{3}(n)]\big{)}$ . Note that this function is continuous but not differentiable, nor absolutely continuous, since it is the integral of a singular measure (the Hausdorff measure of $\mathtt{C}_{\infty}$ ). It is, however, Hölder continuous with exponent $\tau$ . We can also find the range of oscillation of $C_{n}$ . Indeed, it is easy to see that $\limsup_{n\to\infty}C_{n}=1$ (attained for $n=3^{k}$ ) while $\liminf_{n\to\infty}C_{n}=2^{-\tau}\approx 0.646$ (approached for $n=2\cdot 3^{k}-1$ ).

Remark \theRemark.

The function $h_{0}(e^{-3{{}^{u}}})$ is holomorphic in the strip $|\mathrm{Im}\,u|<\frac{\pi}{2\ln 3}$ and periodic under $u\mapsto u+1$ , so we may consider its Fourier coefficients $A_{n}$ . They are related to the Fourier coefficients $a_{n}$ of the measure $d\mu(e^{-3^{u}})$ by the formula

\displaystyle A_{n}=\Gamma(\tau+\tfrac{2\pi in}{\ln 3})a_{n}.

Here and throughout $\Gamma(c)$ denotes the gamma function evaluated at $c\in\mathbb{C}$ (not to be confused with the group $\Gamma$ ). Since $\mu$ has unit volume on the period, we have $|a_{n}|\in O(1)$ , $n\to\infty$ . Hence:

	$\displaystyle\|A_{n}\|\in O\big{(}\|\Gamma(\tau+\tfrac{2\pi in}{\ln 3})\|\big{)}=O(n^{\tau-\frac{1}{2}}e^{-\frac{\pi^{2}n}{\ln 3}}),\ n\to\infty.$
	$\displaystyle\leavevmode\hbox to193.85pt{\vbox to120.13pt{\pgfpicture\makeatletter\hbox{\hskip 96.92265pt\lower-60.06496pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-93.58965pt}{-56.73195pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=113.81102pt]{figs/cantor3c}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.36137pt}{17.66684pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=3$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.$

The numbers $|A_{n}|$ are tiny for large $n$ , see the above loglogplot.

2BII. Generalization to general primes and highest weights

A similar analysis applies for any prime $p$ and rational highest weights $\lambda\in\mathbb{Z}_{p}\cap\mathbb{Q}$ (excluding positive integers, for which $L_{\lambda}$ is finite dimensional). Namely, in this case $\lambda$ has an infinite periodic expansion, which can be computed in the standard way: write $\lambda$ as $q-\frac{m}{n}$ where $0<\frac{m}{n}\leq 1$ and $q\in\mathbb{Z}$ , and pick the minimal $N\in\mathbb{Z}_{>0}$ such that $n$ divides $p^{N}-1$ . Then the repeating string of the expansion of $\lambda$ is the base $p$ expression of the number $r(\lambda):=\frac{m(p^{N}-1)}{n}$ (a string of length $N$ , with zeros at the beginning if needed). If $r(\lambda)=[r_{N-1},\dots,r_{0}]$ , where $r_{j}$ are its base $p$ digits, then

\displaystyle\tau(\lambda)=\frac{1}{N}\sum_{j=0}^{N-1}\log_{p}(r_{j}+1).

The corresponding fractal $\mathtt{C}(\lambda)$ is the set of real numbers which have a base $p$ expansion with $j$ th digit in $\{0,\dots,r_{j\text{ mod }N}\}$ , which has Hausdorff (and Minkowski) dimension $\tau(\lambda)$ . Note that for $\lambda\in\mathbb{Z}_{<0}$ , $\mathtt{C}(\lambda)=\mathbb{R}_{\geq 0}$ (so $\tau(\lambda)=1$ ), while for $\lambda\notin\mathbb{Z}$ we get an actual fractal, i.e. $\tau(\lambda)<1$ is transcendental. For example, for $p=3$ we have $\mathtt{C}(-1/2)=\mathtt{C}_{\infty}$ and $\tau(-1/2)=\log_{3}2$ as explained above.

The deeper analysis of the oscillating behavior of the character of $L_{\lambda}$ and the sequence of Cesáro sums of its coefficients (i.e., the sequence $\dim_{\mathbf{k}}L_{\lambda}[\leq n]$ of dimensions of weight spaces $\leq n$ ) also extends mutatis mutandis to general $p$ and $\lambda$ . In fact, the power behavior of the sequence $\dim_{\mathbf{k}}L_{\lambda}[\leq n]$ is observed for sufficiently generic $\lambda\in\mathbb{Z}_{p}$ . Namely, if $\lambda=\sum_{j=0}^{\infty}r_{j}p^{j}$ and there exists a limit

\displaystyle\tau(\lambda):=\lim_{N\to\infty}\frac{1}{N}\sum_{j=0}^{N-1}\log_{p}(r_{j}+1),

then we have

\displaystyle\lim_{n\to\infty}\frac{\log\dim_{\mathbf{k}}L_{\lambda}[\leq n]}{\ln n}=\tau(\lambda).

This includes rational $\lambda$ , and also if $\lambda$ is chosen randomly (all digits are independent and uniformly distributed) then

\displaystyle\tau(\lambda)=\frac{1}{p}\log_{p}(p!).

For example, for $p\neq 2$ , let $\tau=\log_{p}(p-1)$ . Consider the base $p$ Cantor set sequence:

\displaystyle\mathrm{ca}_{n}^{p}=\begin{cases}1&\text{if the base $p$ expansion of $n$ contains no $1$},\\ 0&\text{otherwise}.\end{cases}

Thus, we have:

\displaystyle p=5\colon\tau\approx 0.861,\quad p=7\colon\tau\approx 0.921,\quad p=11\colon\tau\approx 0.960.

All of these values are transcendental.

Then the analog of Equation 2B.3 for $p=5$ and $p=7$ is:

	$\displaystyle\leavevmode\hbox to201.8pt{\vbox to120.13pt{\pgfpicture\makeatletter\hbox{\hskip 100.8975pt\lower-60.06496pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-97.5645pt}{-56.73195pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=113.81102pt]{figs/cantor5}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.36137pt}{26.20276pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=5$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\leavevmode\hbox to225.64pt{\vbox to120.13pt{\pgfpicture\makeatletter\hbox{\hskip 112.82205pt\lower-60.06496pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-109.48904pt}{-56.73195pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=113.81102pt]{figs/cantor5b}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.36137pt}{26.20276pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=5$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$
	$\displaystyle\leavevmode\hbox to208.3pt{\vbox to120.13pt{\pgfpicture\makeatletter\hbox{\hskip 104.14966pt\lower-60.06496pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-100.81665pt}{-56.73195pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=113.81102pt]{figs/cantor7}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.36137pt}{26.20276pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=7$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\leavevmode\hbox to223.48pt{\vbox to120.13pt{\pgfpicture\makeatletter\hbox{\hskip 111.738pt\lower-60.06496pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-108.405pt}{-56.73195pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=113.81102pt]{figs/cantor7b}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.36137pt}{26.20276pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=7$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.$

2BIII. Generalization to higher rank

The above analysis can also be extended to simply connected simple algebraic groups $G$ of arbitrary rank $r$ . In this case, a weight $\lambda$ is an $r$ -tuple $(\lambda_{1},\dots,\lambda_{r})$ , $\lambda_{j}\in\mathbb{Z}_{p}$ (the coefficients of $\lambda$ with respect to fundamental weights), and a fractal attached to the simple highest weight module $L_{\lambda}$ over the distribution algebra $\mathrm{Dist}(\Gamma)$ can be built in a Euclidean space $\mathrm{Fun}(R_{+},\mathbb{R})$ , where $R_{+}$ is the set of positive roots of $\Gamma$ , for which we choose an ordering. For example, suppose $\lambda=\frac{\mu}{1-p}$ , where $\mu=(\mu_{1},\dots,\mu_{r})$ is an integral weight with $0\leq\mu_{i}\leq p-1$ . Then the fractal $\mathtt{C}(\lambda,B)$ attached to $L_{\lambda}$ depends on a choice of a PBW basis $B$ of $L_{\mu}$ .

Namely, pick a collection $B\subset\mathrm{Fun}(R_{+},[0,p-1])$ such that the set of vectors $\prod_{\tau\in R_{+}}e_{\tau}^{b(\tau)}v_{\mu}$ , $b\in B$ (product in the chosen order) forms a basis of $L_{\mu}$ (where $v_{\mu}\in L_{\mu}$ is the highest weight vector). Then we define $\mathtt{C}(\lambda,B)$ as the set of functions $\phi\colon R_{+}\to\mathbb{R}_{\geq 0}$ which have a base $p$ expansion such that for all $j\in\mathbb{Z}$ , the $j$ th digit $\phi_{j}$ of $\phi$ belongs to $B$ . The Hausdorff (and Minkowski) dimension of $\mathtt{C}(\lambda,B)$ equals

\displaystyle\tau(\lambda)=\log_{p}\dim_{\mathbf{k}}L_{\mu}

for any choice of $B$ . (However, it must be mentioned that $\dim_{\mathbf{k}}L_{\mu}$ is notoriously hard to compute in general.)

Also, $\mathtt{C}(\lambda,B)$ comes equipped with a natural measure $\mu_{B}$ , obtained by suitable rescaling of the counting measures as before (a multiple of the Hausdorff measure). Finally, we have a proper map $\pi\colon\mathtt{C}(\lambda,B)\to\mathbb{R}_{\geq 0}^{r}=\mathbb{R}_{\geq 0}R_{+}$ given by $\pi(\phi)=\sum_{\tau\in R_{+}}\phi(\tau)\tau$ , and the measure $\mu:=\pi_{\ast}\mu_{B}$ does not depend on $B$ , nor on the ordering of positive roots (it expresses the large-scale asymptotics of the character of $L_{\lambda}$ ). This measure completely determines the large-scale behavior of $L_{\lambda}$ .

2BIV. Modular representations and Sierpinski gaskets

Fix a prime $p$ . To give another example, recall Sierpinski’s gasket or triangle, which is Pascal’s triangle modulo $p$ . For $p=2$ we have (we only illustrate cutoffs):

\displaystyle\leavevmode\hbox to149.04pt{\vbox to149.04pt{\pgfpicture\makeatletter\hbox{\hskip 74.51895pt\lower-74.51895pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-71.18594pt}{-71.18594pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=142.26378pt]{figs/striangle1}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.36137pt}{26.20276pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=2$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\quad\leavevmode\hbox to149.04pt{\vbox to149.04pt{\pgfpicture\makeatletter\hbox{\hskip 74.51895pt\lower-74.51895pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-71.18594pt}{-71.18594pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=142.26378pt]{figs/striangle2}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.36137pt}{26.20276pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=2$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.

Sierpinski’s triangle is combinatorially given by the limit of the above pictures, keeping the black boxes and disregarding the white boxes.

This example is related to representation theory of $\Gamma=SL_{2}(\mathbf{k})$ as follows.

For each $N$ let us plot the weights of $L_{n}$ for $n\leq p^{N}-1$ on the line $x+y=n$ on the coordinate plane: the weight $n-2k$ corresponds to the point $(n-k,k)$ . Let us rescale the obtained set by the factor $p^{-N}$ and denote the resulting set by $\Delta_{N}(p)$ . Thus $\Delta_{N}(p)$ is the set of pairs $(x,y)$ , $x=\frac{X}{p^{N}}$ , $y=\frac{Y}{p^{N}}$ where $X,Y$ are $\leq N$ -digit numbers in base $p$ such that there are no carries in computing $X+Y$ (i.e., for all $i$ , $X_{i}+Y_{i}\leq p-1$ ). This shows that $\Delta_{N-1}(p)\subset\Delta_{N}(p)$ .

Let $\Delta_{\infty}(p)=\cup_{N\geq 1}\Delta_{n}(p)$ and $\Delta(p)$ be the closure of $\Delta_{\infty}(p)$ . Hence, $\Delta(p)$ is the compact set of pairs $(x,y)\in\mathbb{R}^{2}$ such that $x=0.x_{1}x_{2}\dots$ , $y=0.y_{1}y_{2}\dots$ in base $p$ , and $x_{i}+y_{i}\leq p-1$ . This set is the Sierpinski triangle from above.

Remark \theRemark.

The set $\Delta_{\infty}(p)$ also has a direct representation-theoretic interpretation. Namely it corresponds to the set of weights in the simple representations of the (infinite type) affine group scheme $(SL_{2})_{\mathrm{perf}}$ , obtained by ‘perfecting’ $SL_{2}$ , see [CW24, §7.1]. It would be interesting to have a similar direct interpretation for $\mathtt{C}_{\infty}$ from Section 2BI. This is less obvious since distribution algebras of perfected group schemes are trivial, see [CW24, Lemma 3.1.2].

It is easy to see that $|\Delta_{N}|=\binom{p}{2}^{N}$ . From this it is not hard to deduce the well-known fact that the Hausdorff dimension of $\Delta(p)$ is the number given by

\displaystyle\tau=\log_{p}\binom{p}{2}=1+\log_{p}\frac{p+1}{2}=1+\frac{\ln\frac{p+1}{2}}{\ln p},\quad\text{e.g. }\tau\approx 1.631\text{ for $p=3$}.

This number is transcendental, and can also be seen at the level of the generating function given by

\displaystyle f(z):=\sum_{n=0}^{\infty}(\dim_{\mathbf{k}}L_{n})z^{n}.

Steinberg’s tensor product theorem yields that $f$ is a 2-Mahler equation of degree $p$ :

	$\displaystyle f(z)$	$\displaystyle=\prod_{m=0}^{\infty}\frac{1-(p+1)z^{p^{m+1}}+pz^{(p+1)p^{m}}}{(1-z^{p^{m}})^{2}}=(1+2z+\dots+pz^{p-1})\cdot f(z^{p})$
		$\displaystyle=\frac{1-(p+1)z^{p}+pz^{p+1}}{(1-z)^{2}}\cdot f(z^{p}).$

We have $(1+2z+\dots+pz^{p-1})|_{z=1}=\binom{p}{2}$ , so we have for some $C>1$

\displaystyle f\asymp(1-z)^{-\tau}\colon\quad C^{-1}\cdot(1-z)^{-\tau}\leq f(z)\leq C\cdot(1-z)^{-\tau}.

This implies that for some $\widetilde{C}>1$ , we have, for large enough $n$ :

\displaystyle\sum_{j=0}^{n}\dim_{\mathbf{k}}L_{j}\asymp n^{\tau}\colon\quad\widetilde{C}^{-1}\cdot n^{\tau}\leq\sum_{j=0}^{n}\dim_{\mathbf{k}}L_{j}\leq\widetilde{C}\cdot n^{\tau}.

This is illustrated in Equation 2B.4 below. A more detailed asymptotics of $f(z)$ can be obtained as follows. Note that

\displaystyle(1-z)^{\tau}\cdot f(z)=\prod_{m=0}^{\infty}\frac{1+2z^{p^{m}}+\dots+pz^{(p-1)p^{m}}}{(1+z^{p^{m}}+\dots+z^{(p-1)p^{m}})^{\tau}}.

Thus, we see that

\displaystyle\lim_{m\to\infty}(1-z^{p^{-m}})^{\tau}\cdot f(z^{p^{-m}})=g_{p}(z),

where (note that the product below is absolutely convergent)

\displaystyle g_{p}(z):=\prod_{m=-\infty}^{\infty}\frac{1+2z^{p^{m}}+\dots+pz^{(p-1)p^{m}}}{(1+z^{p^{m}}+\dots+z^{(p-1)p^{m}})^{\tau}}

is a periodic function in the sense that $g_{p}(z)=g_{p}(z^{p})$ . This implies that the sequence $L(n)=n^{-\tau}\sum_{j=0}^{n}\dim_{\mathbf{k}}L_{j}$ also exhibits oscillatory behavior:

(2B.4)

\displaystyle\leavevmode\hbox to201.07pt{\vbox to120.13pt{\pgfpicture\makeatletter\hbox{\hskip 100.53615pt\lower-60.06496pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-97.20314pt}{-56.73195pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=113.81102pt]{figs/striangle3a}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.36137pt}{31.89322pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=3$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\quad\leavevmode\hbox to188.06pt{\vbox to120.13pt{\pgfpicture\makeatletter\hbox{\hskip 94.03186pt\lower-60.06496pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-90.69885pt}{-56.73195pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=113.81102pt]{figs/striangle3}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.36137pt}{31.89322pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=3$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.

Let $\mu_{p}$ be the direct image of the (suitably normalized) Hausdorff measure on $\Delta_{\infty}(p)$ under the map $(x,y)\mapsto x+y$ . Then the analytic function $\mathtt{h}_{p0}(v):=v^{-\tau}h_{p0}(e^{-v})$ on $(0,\infty)$ is the Laplace transform of $\mu_{p}$ :

	$\displaystyle\mathtt{h}_{p0}(v)=\int_{0}^{\infty}e^{-tv}d\mu_{p}(t),$
	$\displaystyle h_{p0}(x)=\ln(x^{-1})^{\tau}\int_{0}^{\infty}x^{t}d\mu_{p}(t).$

Similarly, the sequence $C_{n}(p):=n^{-\tau}\sum_{j=0}^{n}\dim_{\mathbf{k}}L_{j}$ for large $n$ behaves as follows:

\displaystyle\lim_{m\to\infty}C_{[p^{m}w]}=w^{-\tau}\mu_{p}([0,w]),

Thus, this sequence approaches the periodic function $\log_{p}(n)^{-\tau}\mu_{p}([0,\log_{p}(n)])$ at infinity. As before, this function is continuous but not differentiable, nor absolutely continuous.

2BV. Lengths of tensor powers

Recall that $\ell({}_{-})$ denotes the length of a representation. Finally let us discuss the fractal behavior of the integer sequence $l_{n}=\ell(V^{\otimes n})$ where $V\cong\mathbf{k}^{2}$ is the vector representation of $\Gamma$ . We will not explicitly consider any fractals in this example, however the origin of what we, based on the previous examples, call ‘fractal behavior’ is still found in the same principles, such as Steinberg’s tensor product theorem.

$p=2$ . For simplicity consider the case $p=2$ first. In this case one can show that the generating function

\displaystyle L(z):=\sum_{n\geq 0}l_{n}z^{-n-1}

is holomorphic for $|z|>2$ and satisfies the functional equation

\displaystyle L(z)=(1+z)\cdot L(z^{2}-2).

In particular, this shows that $l_{2k-1}=l_{2k}$ for all $k\in\mathbb{Z}_{>0}$ . This is not a Mahler equation, but it turns into one under a simple change of variable. Namely, setting $z=w+w^{-1}$ and $wh(w)=L(z)$ , we have

\displaystyle h(w)=(1+w+w^{2})\cdot h(w^{2}),

i.e., $h$ is a 2-Mahler function of degree $2$ .

One gets

\displaystyle h(w)=\prod_{n\geq 0}(1+w^{2^{n}}+w^{2\cdot 2^{n}})\text{ for }|w|<1.

Using this, we can compute as previously the asymptotics of $h$ as $w\uparrow 1$ . Set

\displaystyle\tau=\log_{2}3\approx 1.585,

which is transcendental. We then have, again by the theory of Mahler functions,

\displaystyle C_{1}\cdot(1-w)^{-\tau}\big{(}1+o(1)\big{)}\leq h(w)\leq C_{2}\cdot(1-w)^{-\tau}\big{(}1+o(1)\big{)}\text{ for }w\uparrow 1,

for some $0<C_{1}<C_{2}$ with $C_{1},C_{2}\in\mathbb{R}$ . Hence, we get

\displaystyle C_{1}\cdot(z-2)^{-\tau/2}\big{(}1+o(1)\big{)}\leq L(z)\leq C_{2}\cdot(z-2)^{-\tau/2}\big{(}1+o(1)\big{)}\text{ for }z\downarrow 2.

So from the Tauberian theory, cf. Section 6A, it follows that

\displaystyle\frac{2^{1-\frac{\tau}{2}}C_{1}}{\Gamma(\frac{\tau}{2}+1)}\cdot n^{\frac{\tau}{2}}\big{(}1+o(1)\big{)}\leq\sum_{j=0}^{n}\frac{l_{j}}{2^{j}}\leq\frac{2^{1-\frac{\tau}{2}}C_{2}}{\Gamma(\frac{\tau}{2}+1)}\cdot n^{\frac{\tau}{2}}\big{(}1+o(1)\big{)}\text{ for }n\to\infty.

Or, equivalently,

\displaystyle\frac{2C_{1}}{3\Gamma(\frac{\tau}{2}+1)}\cdot n^{\frac{\tau}{2}}\big{(}1+o(1)\big{)}\leq\sum_{k=1}^{n}\frac{l_{2k}}{2^{2k}}\leq\frac{2C_{2}}{3\Gamma(\frac{\tau}{2}+1)}\cdot n^{\frac{\tau}{2}}\big{(}1+o(1)\big{)}\text{ for }n\to\infty.

As previously, the behavior of $h(w)$ as $w\uparrow 1$ can be analyzed as follows. We have

\displaystyle\lim_{k\to\infty}\ln(w^{-2^{-k}})^{\tau}h(w^{2^{-k}})=\ln(w^{-1})^{\tau}\lim_{k\to\infty}3^{-k}\prod_{m=-k}^{\infty}(1+w^{2^{m}}+w^{2\cdot 2^{m}})=\mathtt{h}_{0}(w),

where (for $\theta(m)$ as in Section 2BI above)

\displaystyle\mathtt{h}_{0}(w):=\ln(w^{-1})^{\tau}\prod_{m=-\infty}^{\infty}\frac{1+w^{2^{m}}+w^{2\cdot 2^{m}}}{3^{\theta(m)}}.

This is a periodic function in the sense that $\mathtt{h}_{0}(w)=\mathtt{h}_{0}(w^{2})$ , and the function $\ln(w^{-1})^{\tau}h(w)$ asymptotically approaches the periodic function $\mathtt{h}_{0}(w)$ as $w\uparrow 1$ . Writing $w=e^{-v}$ , we obtain that the function $v^{\tau}h(e^{-v})$ approaches $\mathtt{h}_{0}(e^{-v})$ as $v\downarrow 0$ , i.e.,

\displaystyle\lim_{k\to\infty}(2^{-k}v)^{\tau}h(e^{-2^{-k}v})=\mathtt{h}_{0}(e^{-v}).

Or, equivalently,

\displaystyle\lim_{k\to\infty}(2^{-k}v)^{\tau}L(2\cosh 2^{-k}v)=\mathtt{h}_{0}(e^{-v}).

But

\displaystyle 2\cosh 2^{-k}v=2e^{2^{-2k-1}v^{2}}+O(2^{-4k})\text{ for }k\to\infty,

so we get

\displaystyle\lim_{k\to\infty}(2^{-k}v)^{\tau}L(2e^{2^{-2k-1}v^{2}})=\mathtt{h}_{0}(e^{-v}).

Thus, setting $x:=e^{v^{2}/2}$ , we obtain

\displaystyle\lim_{k\to\infty}(\ln x^{4^{-k}})^{\tau/2}L(2x^{4^{-k}})=\widetilde{h_{0}}(x):=2^{-\tau}\mathtt{h}_{0}(e^{-\sqrt{2\ln x}}).

Hence, the function

\displaystyle\widetilde{h}_{0}(e^{\frac{1}{2}4^{u}})=2^{-\tau}\mathtt{h}_{0}(e^{-2^{u}})

is periodic with period $1$ and analytic for $|\mathrm{Im}\,u|\leq\frac{\pi}{\ln 4}$ , i.e., the strip of holomorphy is twice as wide as in previous examples. This happens because the role of the prime $p$ in this example is played by the number $4$ (rather than $2$ ), i.e., as $z\downarrow 2$ , the function $L(z)$ behaves as a Mahler function of degree four, rather than a Mahler function of degree two. As we will see, this will lead to much greater regularity of the coefficient sequence $l_{2n}$ .

We would now like to understand the asymptotics of the sequence $l_{2n}$ in more detail. If $l_{2k}$ was known to behave sufficiently regularly, we could read off its asymptotics from the asymptotics of the Cesáro sums $\sum_{k=1}^{2n}\frac{l_{2k}}{2^{2k}}$ by Abel resummation:

\displaystyle\frac{2C_{1}}{3\Gamma(\frac{\tau}{2})}\cdot n^{-1+\frac{\tau}{2}}\cdot 2^{2n}\big{(}1+o(1)\big{)}\leq l_{2n}\leq\frac{2C_{2}}{3\Gamma(\frac{\tau}{2})}\cdot n^{-1+\frac{\tau}{2}}\cdot 2^{2n}\big{(}1+o(1)\big{)}\text{ for }n\to\infty.

The required regularity is guaranteed by the following lemma, showing that the sequence $\frac{l_{2n}}{2^{2n}}$ is decreasing.

Lemma \theLemma.

We have $4l_{n}\geq l_{n+2}$ .

Proof.

Let $Q(z):=-(1-4z^{-2})L(z)=\sum_{n\geq 0}(4l_{n-2}-l_{n})z^{-n-1}$ , where we agree that $l_{-1}=l_{-2}:=0$ . Then the functional equation for $L$ implies that

\displaystyle Q(z)=z^{-4}(1+z)(z^{2}-2)^{2}Q(z^{2}-2).

So the coefficients $c_{n}=4l_{n}-l_{n+2}$ of $Q(z)$ satisfy the recursion

\displaystyle\sum_{n\geq 0}c_{n}z^{-n-3}=z^{-3}+z^{-4}+z^{-5}(1+z^{-1})\sum_{n\geq 0}\frac{c_{n}z^{-2n}}{(1-2z^{-2})^{n+1}}.

This shows that $4l_{n}-l_{n+2}\geq 0$ for all $n\geq 0$ , which implies the statement. ∎

From this it follows (with some work) that

\displaystyle\lim_{k\to\infty}2^{-2\cdot[4^{k}y]}[4^{k}y]^{1-\tau/2}l_{2\cdot[4^{k}y]}=\phi(\log_{4}y),

where $\phi(y)$ is a periodic function in the sense that $\phi(y)=\phi(4y)$ (period doubling with respect to the previous examples). Thus, $l_{2n}$ behaves roughly like $(2n)^{0.21}\cdot 2^{2n}$ with $1-\frac{1}{2}\log_{2}3\approx 0.21$ (as expected, growing faster than in characteristic zero, where it is $(2n)^{-1/2}\cdot 2^{2n}$ , see Section 1B). Here is the plot (for $l_{n}$ ; note that $l_{2k+1}=l_{2k}$ ):

\displaystyle\leavevmode\hbox to214.8pt{\vbox to134.58pt{\pgfpicture\makeatletter\hbox{\hskip 107.40181pt\lower-67.29195pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-104.0688pt}{-63.95894pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=128.0374pt]{figs/lengtha}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.36137pt}{31.89322pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=2$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\quad\leavevmode\hbox to217.69pt{\vbox to134.58pt{\pgfpicture\makeatletter\hbox{\hskip 108.8472pt\lower-67.29195pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-105.51419pt}{-63.95894pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=128.0374pt]{figs/lengthb}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.36137pt}{31.89322pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=2$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.

To see the oscillation, we zoom into $l_{2n}/((2n)^{0.21}\cdot 2^{2n})$ :

\displaystyle\leavevmode\hbox to203.24pt{\vbox to134.58pt{\pgfpicture\makeatletter\hbox{\hskip 101.62021pt\lower-67.29195pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-98.2872pt}{-63.95894pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=128.0374pt]{figs/lengthc}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-28.09596pt}{31.89322pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=2$, even $n$ zoom}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.

As before, the Fourier coefficients $a_{n}$ of the 1-periodic function $\phi(4^{u})$ are related to the Fourier coefficients $A_{n}$ of the function $2^{-\tau}\mathtt{h}_{0}(e^{-2^{u}})$ by the formula

\displaystyle A_{n}=\Gamma(\tfrac{\tau}{2}+1+\tfrac{2\pi in}{\ln 4})a_{n}.

So since $\mathtt{h}_{0}(e^{-2^{u}})$ is analytic in the strip $|\mathrm{Im}\,u|\leq\frac{\pi}{\ln 4}$ , we have $|A_{n}|=O(e^{-(\frac{2\pi^{2}}{\ln 4}-\varepsilon)|n|})$ for all $\varepsilon>0$ , hence

\displaystyle|a_{n}|=O(e^{-(\frac{\pi^{2}}{\ln 4}-\varepsilon)|n|})\text{ for }|n|\to\infty.

Thus, the function $\phi(4^{u})$ is analytic in the strip $|\mathrm{Im}\,u|\leq\frac{\pi}{2\ln 4}$ .

This is a significant difference from the previous examples, where the analogous function was not even absolutely continuous (nor differentiable). This happens due to presence of the change of variable $z=w+w^{-1}$ which has a quadratic branch point at $w=1$ and thus causes doubling of periods and widths of strips of holomorphy.

General $p$ . The analysis is essentially the same as for $p=2$ , and we will omit a discussion. Let us simply point out that the generating function satisfies

\displaystyle L(w+w^{-1})=w^{1-p}(1+w+w^{2}+\dots+w^{2p-2})\cdot L(w^{p}+w^{-p})

which gives

\displaystyle h(w)=(1+w+w^{2}+\dots+w^{2p-2})\cdot h(w^{p}).

Hence, we have again a 2-Mahler function of degree $p$ . From this we get

\displaystyle\tau=\log_{p}(2p-1)

as the exponent of the subexponential factor.

2BVI. Conclusion and goal

The primary objective of this paper is to study a similar but more complicated problem of precisely estimating the number of indecomposable summands of $V^{\otimes n}$ . Unlike the study of the length, which grows faster than the number of indecomposable summands due to the non semisimple nature of the category of representations of $SL_{2}(\mathbf{k})$ , this problem introduces subtleties. For example, since the 2-Mahler equation of degree $p$ it leads to is inhomogeneous, the generating function $h(w)$ is not a single product but rather a sum of products. Still, the asymptotic behavior of the sequences and functions we consider ends up being very similar to the above examples. There will be an especial similarity with the example in this subsection (e.g. length of $V^{\otimes n}$ ). Namely, we also observe doubling of periods and strip widths, resulting in a very high degree of regularity of the sequence of interest.

We will however abstain from exploring specific fractals within this context.

3. Generating function

We fix a prime $p$ . Our first goal is to explicitly describe the generating function for the sequence $b_{n}$ as in Equation 1B.1. All objects will depend on $p$ , but to keep notation light we usually omit $p$ from notation.

3A. The function $F$

Let $w$ and $z$ be formal variables. By expansion, we have a ring inclusion

\displaystyle\mathbb{Q}[[z^{-1}]]\hookrightarrow\mathbb{Q}[[w]],\text{ corresponding to $z\mapsto w+w^{-1}$}.

Moreover, the above restricts to inclusions

\displaystyle\mathbb{Z}[[z^{-1}]]\hookrightarrow\mathbb{Z}[[w]]\quad\text{and}\quad\mathbb{Q}(z)=\mathbb{Q}(z^{-1})\hookrightarrow\mathbb{Q}(w).

The image of the latter inclusion consists precisely of those rational functions in $w$ invariant under $w\leftrightarrow w^{-1}$ . We will use this several times tacitly below.

We are now ready to study the generating function for the sequence $b_{n}$ . It is a bit more convenient to shift the generating function for $b_{n}$ as follows. Let

\displaystyle H(z):=H_{p}(z):=\sum_{n\geq 0}b_{n}z^{-n-1}\in\mathbb{Z}[[z^{-1}]].

We will also regard $H(z)$ as a holomorphic function with domain $|z|>2$ which vanishes at infinity.

It will be convenient for formal manipulations to focus on $H(w+w^{-1})$ . We therefore set

\displaystyle F(w):=F_{p}(w):=\sum_{n\geq 0}b_{n}(w+w^{-1})^{-n-1}\in\mathbb{Z}[[w]].

Of course, by construction, we can also interpret $F(w)$ as a holomorphic function (the singularity of this function that will be important later is now at $w=1$ ) on

(3A.1)

\displaystyle\mathbb{C}\supset\Omega:=\{w\in\mathbb{C}\mid|w+w^{-1}|>2\}.\quad\text{Plot of $\Omega$}\colon\leavevmode\hbox to118.68pt{\vbox to120.13pt{\pgfpicture\makeatletter\hbox{\hskip 59.34225pt\lower-60.06496pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-56.00925pt}{-56.73195pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=113.81102pt]{figs/region}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.

Indeed, direct manipulation shows that $\Omega$ is the set of complex numbers $x+iy$ with

\displaystyle\big{(}x^{2}+(y-1)^{2}-2\big{)}\big{(}x^{2}+(y+1)^{2}-2\big{)}>0,

yielding the intersection and joint exterior of two disks, as displayed in Equation 3A.1.

Our main result of this section is:

Theorem \theTheorem.

We have the following explicit formula for $F(w)$ :

\displaystyle F(w)=\sum_{k=0}^{\infty}\frac{w^{p^{k}}(1-w^{p^{k}(p-1)})}{(1-w^{p^{k}})(1+w^{p^{k+1}})}\prod_{j=0}^{k-1}\frac{1-w^{(p+1)p^{j}}}{(1-w^{p^{j}})(1+w^{p^{j+1}})}.

Remark \theRemark.

As pointed out to us by Henning Haahr Andersen, a finer version (with a quite different formulation) of Section 3A has appeared in [Erd95].

The remainder of this section is devoted to the proof of Section 3A.

Example \theExample.

If $p=2$ , then the expression in Section 3A simplifies to

\displaystyle F(w)=\sum_{k=0}^{\infty}\frac{1}{w^{2^{k}}+w^{-2^{k}}}\prod_{j=0}^{k-1}\left(1+\frac{1}{w^{2^{j}}+w^{-2^{j}}}\right),

as one can directly verify.

Example \theExample.

We can also consider the generating function for a field of characteristic zero for the numbers $b_{n}$ , see Section 1B. Then we find

	$\displaystyle H_{\infty}(z)=$	$\displaystyle\sum_{n\geq 0}\binom{n}{\lfloor n/2\rfloor}z^{-n-1}=z^{-1}+z^{-2}+3z^{-3}+6z^{-5}+\dots$
	$\displaystyle=$	$\displaystyle\frac{1}{2}\left(\sqrt{\frac{z+2}{z-2}}-1\right).$

This is well-known and can be derived from [OEI23, A001405].

3B. Recursion relations

Recall the category of tilting representations for $SL_{2}(\mathbf{k})$ , see e.g. [Don93] and [Rin91] (additional details can be found in [AST18]), or, using the identification with the Temperley–Lieb calculus, [And19] or [TW21]. All the terminology and facts about tilting representations that we use below can be found or derived from these references. We denote the category by $\mathrm{Tilt}\big{(}SL_{2}(\mathbf{k})\big{)}$ .

We identify the weight lattice of $SL_{2}(\mathbf{k})$ with $\mathbb{Z}$ and the dominant weights with $\mathbb{Z}_{\geq 0}$ . Let $K=K_{p}$ be the split Grothendieck ring of $\mathrm{Tilt}\big{(}SL_{2}(\mathbf{k})\big{)}$ . Then $K$ has a $\mathbb{Z}$ -basis $\{[\mathtt{T}_{i}]\mid i\in\mathbb{Z}_{\geq 0}\}$ , where $\mathtt{T}_{i}$ is the indecomposable tilting representation of highest weight $i$ . We then have ring isomorphisms

\displaystyle\mathbb{Z}[u,u^{-1}]^{s}\xleftarrow{\,\mathrm{ch}\,}K\xrightarrow{[V]\mapsto x}\mathbb{Z}[x],

where $\mathbb{Z}[u,u^{-1}]^{s}$ is the subring of Laurent polynomials symmetric under $u\leftrightarrow u^{-1}$ , and $ch$ associates the formal character to a representation. In particular, the composed isomorphism identifies $x$ and $u+u^{-1}$ . We will freely use both isomorphisms and switch notation accordingly.

Let $\nu=\nu_{p}\colon\mathbb{Z}[u,u^{-1}]^{s}\simeq\mathbb{Z}[x]\simeq K\to\mathbb{Z}$ be the group homomorphism

\displaystyle\nu\colon K\to\mathbb{Z},\;[\mathtt{T}_{i}]\mapsto 1,\text{ for all $i\in\mathbb{Z}_{\geq 0}$}.

Hence, we have $\nu(x^{n})=b_{n}$ .

By Donkin’s tensor product formula, for $0\leq k<p$ the operation $\mathtt{T}\mapsto\mathtt{T}^{(1)}\otimes\mathtt{T}_{p-1+k}$ maps indecomposable tilting representations to indecomposable tilting representations; concretely:

\displaystyle\mathtt{T}_{i}^{(1)}\otimes\mathtt{T}_{p-1+k}\;\simeq\;\mathtt{T}_{pi+p-1+k}.

Here $\mathtt{T}^{(1)}$ is the Frobenius twist of $\mathtt{T}$ . In particular, we have

\displaystyle ch(\mathtt{T}^{(1)})=\mathbb{P}\big{(}ch(\mathtt{T})\big{)},

where $\mathbb{P}=\mathbb{P}_{p}$ is the endomorphism of $\mathbb{Z}[u,u^{-1}]$ determined by $u\mapsto u^{p}$ . Consequently, we have

(3B.1)

\displaystyle\nu=\nu\big{(}\mathbb{P}({}_{-})\,ch(\mathtt{T}_{p-1+k})\big{)},

for all $0\leq k<p$ .

We will interpret (3B.1) as recursion relations to compute $b_{n}=\nu(x^{n})$ . To this end, we need to know the following few characters: it is well-known and easy to compute that

\displaystyle ch(\mathtt{T}_{p-1})=\frac{u^{p}-u^{-p}}{u-u^{-1}}

and, for $0<k<p$ ,

(3B.2)

\displaystyle ch(\mathtt{T}_{p-1+k})=(u^{k}+u^{-k})\frac{u^{p}-u^{-p}}{u-u^{-1}}.

We can reformulate this in terms of the Chebyshev polynomials of the first kind, $T_{n}$ (not to be confused with the notation for tilting modules) and of the second kind, $U_{n}$ . Indeed, they can be defined via

(3B.3)

\displaystyle T_{n}\Big{(}\frac{w+w^{-1}}{2}\Big{)}=\frac{w^{n}+w^{-n}}{2},\quad\text{and}\quad U_{n}\Big{(}\frac{w+w^{-1}}{2}\Big{)}=\frac{w^{n+1}-w^{-n-1}}{w-w^{-1}}.

We then find:

	$\displaystyle b_{n}=\nu(x^{n})=$	$\displaystyle\nu\big{(}2^{n}T_{p}(x/2)^{n}U_{p-1}(x/2)\big{)}$
	$\displaystyle=$	$\displaystyle\nu\big{(}2^{n+1}T_{p}(x/2)^{n}T_{k}(x/2)U_{p-1}(x/2)\big{)},\quad\text{for $0<k<p$.}$

Example \theExample.

For $p=2$ , the two equations become

\displaystyle\nu\big{(}x(x^{2}-2)^{n}\big{)}=\nu(x^{n})\quad\text{and}\quad\nu\big{(}x^{2}(x^{2}-2)^{n}\big{)}=\nu(x^{n}).

The first relation allows one to compute $b_{2n+1}$ in terms of lower cases and the second relation allows one to compute $b_{2n+2}$ . Concretely, one obtains

\displaystyle b_{2n-1}=b_{2n}=\sum_{k=0}^{n-1}\binom{n-1}{k}2^{n-1-k}b_{k}.

This gives a very efficient way of computing the numbers $b_{2n-1}=b_{2n}$ .

Let $\mathbb{A}=\mathbb{A}_{p}$ be the group endomorphism of $\mathbb{Z}[[w]]$ given by

(3B.4)

\displaystyle\mathbb{A}(r)(w)=\frac{1}{p}\sum_{j=0}^{p-1}r(\zeta^{j}w),\quad\text{for }\;r\in\mathbb{Z}[[w]],

where $\zeta=e^{2\pi i/p}$ is a primitive complex $p$ th root of unity. In particular, we have

\displaystyle\mathbb{A}(w^{m})=\begin{cases}w^{m}&\text{if $p$ divides $m$},\\ 0&\text{otherwise}.\end{cases}

Lemma \theLemma.

The function $F(w)$ satisfies

	$\displaystyle F(w^{p})$	$\displaystyle=\mathbb{A}\big{(}F(w)\big{)},\quad\text{and}$
	$\displaystyle F(w^{p})+1$	$\displaystyle=\mathbb{A}\left((w^{k}+w^{-k})F(w)\right),\text{ for }1\leq k\leq p-1.$

Proof.

For convenience, we extend the definition of $\nu\colon\mathbb{Z}[u,u^{-1}]^{s}=K\to\mathbb{Z}$ to

\displaystyle\nu\colon K[[w]]\to\mathbb{Z}[[w]],

$\mathbb{Z}[[w]]$ -linearly. Then we can write

\displaystyle F(w)=\nu\left(\sum_{n\geq 0}\frac{(u+u^{-1})^{n}}{(w+w^{-1})^{n+1}}\right)=\nu\left(\frac{1}{w+w^{-1}-u-u^{-1}}\right).

Using (3B.1) and (3B.2), we can then conclude that

(3B.5)

\displaystyle F(w^{p})=\nu\left(\frac{u^{p}-u^{-p}}{u-u^{-1}}\frac{R_{k}}{w^{p}-w^{-p}-u^{p}-u^{-p}}\right),\text{ for }0\leq k<p,

where $R_{0}=1$ and $R_{k}=u^{k}+u^{-k}$ for $i>0$ . We now complete the proof of the first displayed equation, corresponding to $k=0$ , the other cases being similar.

Writing

\displaystyle\frac{w}{w-u}=-\sum_{i>0}\frac{w^{i}}{u^{i}}

we find that

\displaystyle\mathbb{A}\left(\frac{w}{w-u}\right)=\frac{w^{p}}{w^{p}-u^{p}}.

Consequently, we find

	$\displaystyle\mathbb{A}\left(\frac{1}{w+w^{-1}-u-u^{-1}}\right)=$	$\displaystyle\mathbb{A}\left(\frac{w}{u-u^{-1}}\left(\frac{1}{w-u}-\frac{1}{w-u^{-1}}\right)\right)$
	$\displaystyle=$	$\displaystyle\frac{w^{p}}{u-u^{-1}}\left(\frac{1}{w^{p}-u^{p}}-\frac{1}{w^{p}-u^{-p}}\right)$
	$\displaystyle=$	$\displaystyle\frac{u^{p}-u^{-p}}{u-u^{-1}}\frac{1}{w-w^{-p}-u^{p}-u^{-p}}.$

Comparing this expression with (3B.5) indeed proves the first equation of the lemma. ∎

3C. Functional equation

The following proposition shows that $F(w)$ is a 2-Mahler function of degree $p$ .

Proposition \theProposition.

We have

\displaystyle F(w)=d_{p}(w+w^{-1})+\left(1+d_{p}(w+w^{-1})\right)\cdot F(w^{p}),

where $d_{p}$ is explicitly given by

\displaystyle d_{p}(w+w^{-1})=\frac{w(1-w^{p-1})}{(1-w)(1+w^{p})}.

For $p>2$ , $d_{p}(z)$ is also determined as the rational function in $z^{-1}$ such that $zd_{p}(z)$ is the Padé approximant of order $[\frac{p-1}{2}/\frac{p-1}{2}]$ of $\frac{H_{\infty}(z)z^{p+1}}{H_{\infty}(z)z^{2}+z^{p}}$ .

Example \theExample.

Consider the case $p=2$ . The two equations in Section 3B are

\displaystyle F(w^{2})=\frac{1}{2}\big{(}F(w)+F(-w)\big{)}\quad\text{and}\quad F(w^{2})+1=\frac{w+w^{-1}}{2}\big{(}F(w)-F(-w)\big{)}.

Taking a linear combination of the two equations neutralising the terms in $F(-w)$ yields

\displaystyle\underbrace{(w+w^{-1})}_{=d_{2}(w+w^{-1})^{-1}}F(w)=1+\underbrace{(1+w+w^{-1})}_{=d_{2}(w+w^{-1})^{-1}+1}F(w^{2}).

This is equivalent to the functional equation in Section 3C specialized at $p=2$ . The proof of the case $p>2$ given below is a refinement of this argument.

Proof of Section 3C.

We can obtain the functional equation by taking an appropriate linear combination of the $p$ equalities in Section 3B. Indeed, we use the interpretation of $\mathbb{A}$ in (3B.4), and we will sum with appropriate $w$ -dependent coefficients to make all terms containing $F(\zeta^{k}w)$ for $k>0$ cancel. We can start by adding up the equalities in Section 3B for $1\leq k\leq p-1$ with coefficients $\gamma_{k}(w)\in\mathbb{Q}((w))$ so that the resulting coefficients of $F(\zeta^{k}w)$ yield a value $-C(w)\in\mathbb{Q}((w))$ independent of $1\leq k\leq p-1$ .

So we must have

(3C.1)

\displaystyle\sum_{k=1}^{p-1}(\zeta^{kj}w^{k}+\zeta^{-kj}w^{-k})\gamma_{k}(w)=-C(w)

for all $j\in[1,p-1]$ . Consider

\displaystyle L_{w}(t):=\sum_{k=1}^{p-1}(t^{k}+t^{-k})\gamma_{k}(w)+C(w)\in\mathbb{Q}((w))[t,t^{-1}].

By (3C.1) and its complex conjugate, $L_{w}(\zeta^{j}w)=L_{w}(\zeta^{j}w^{-1})=0$ , for all $j\in[1,p-1]$ . It follows that, up to scaling,

	$\displaystyle L_{w}(t)=t^{1-p}\prod_{j=1}^{p-1}(t-\zeta^{j}w)(t-\zeta^{-j}w)=t^{1-p}\frac{(t^{p}-w^{p})(t^{p}-w^{-p})}{(t-w)(t-w^{-1})}=$
	$\displaystyle t^{1-p}\left(\sum_{j=0}^{p-1}t^{j}w^{-j}\right)\left(\sum_{i=0}^{p-1}t^{i}w^{i}\right)=\sum_{i,j=0}^{p-1}t^{i+j-p+1}w^{i-j}.$

It follows that, setting $[j]_{w}=\frac{w^{j}-w^{-j}}{w-w^{-1}}\in\mathbb{Z}((w))$ , we can choose:

\displaystyle\gamma_{k}(w)=\sum_{i=0}^{p-1-k}w^{2i+k+1-p}=[p-k]_{w},\quad C(w)=[p]_{w}.

Summing over the $p-1$ equations as intended we thus get

\displaystyle p\sum_{j=1}^{p-1}[j]_{w}\big{(}F(w^{p})+1\big{)}=\left(\sum_{j=1}^{p-1}[j]_{w}(w^{p-j}+w^{j-p})\right)F(w)-[p]_{w}\big{(}F(\zeta w)+\cdots+F(\zeta^{p-1}w)\big{)}.

So multiplying the first equation, $F(w^{p})=\mathbb{A}\big{(}F(w)\big{)}$ , by $p[p]_{w}$ , and adding it to the above equation, we obtain

\displaystyle p[p]_{w}F(w^{p})+p\sum_{j=1}^{p-1}[j]_{w}\big{(}F(w^{p})+1\big{)}=\left([p]_{w}+\sum_{j=1}^{p-1}[j]_{w}(w^{p-j}+w^{j-p})\right)F(w).

This can be written as

\displaystyle\sum_{j=1}^{p}(w^{j}-w^{-j})F(w^{p})+\sum_{j=1}^{p-1}(w^{j}-w^{-j})=(w^{p}-w^{-p})F(w),

subsequently as

\displaystyle\left(\frac{1-w^{p+1}}{1-w}-\frac{1-w^{-p-1}}{1-w^{-1}}\right)F(w^{p})+\left(\frac{1-w^{p}}{1-w}-\frac{1-w^{-p}}{1-w^{-1}}\right)=(w^{p}-w^{-p})F(w),

and finally as the expression in the theorem.

The values $b_{n}$ for $n<p$ are equal to the corresponding numbers in characteristic zero. Consequently, we have

\displaystyle\frac{1}{H_{p}(z)}=\frac{1}{H_{\infty}(z)}+O(z^{1-p}).

If $T_{n}$ is the Chebyshev polynomial of the first kind, then we set

\displaystyle P_{p}(x):=2T_{p}(x/2).

In other words:

\displaystyle P_{p}(w+w^{-1})=w^{p}+w^{-p}.

The functional equation then implies

\displaystyle H_{p}(z)=d_{p}(z)+\big{(}1+d_{p}(z)\big{)}\cdot H_{p}\big{(}P_{p}(z)\big{)}.

Since $P_{p}(z)$ is of degree $p$ (as a polynomial in $z$ ), we have (as functions in $z^{-1}$ )

\displaystyle H_{p}\big{(}P_{p}(z)\big{)}=z^{-p}+O(z^{-p-2}).

Rewriting the functional equation, we thus find

\displaystyle d_{p}(z)^{-1}=\frac{1+H_{p}\big{(}P_{p}(z)\big{)}}{H_{p}(z)-H_{p}\big{(}P_{p}(z)\big{)}}=\frac{1}{H_{\infty}(z)}+z^{2-p}+O(z^{1-p}).

It thus follows that $z^{-1}d_{p}(z)^{-1}$ is the Padé approximant of order $[a/b]$ of

\displaystyle\frac{z^{-1}}{H_{\infty}(z)}+z^{1-p}

if $a+b<p$ and $z^{-1}d_{p}(z)$ is a rational function in $z^{-1}$ of degree $(a,b)$ .

It remains to verify the final statement. For $p>2$ consider therefore the expansion

	$\displaystyle d_{p}(w+w^{-1})^{-1}=$	$\displaystyle\frac{1}{1+w+w^{2}+\cdots+w^{p-2}}\cdot\frac{1+w^{p}}{w}$
	$\displaystyle=$	$\displaystyle\frac{(1+w)w^{\frac{p-3}{2}}}{1+w+w^{2}+\cdots+w^{p-2}}\cdot\frac{1+w^{p}}{w^{\frac{p-1}{2}}(1+w)}.$

The left factor on the last line is the inverse of a polynomial in $w+w^{-1}$ of degree $(p-3)/2$ , while the right factor is a polynomial in $w+w^{-1}$ of degree $(p-1)/2$ . In particular, $z^{-1}d_{p}(z)^{-1}$ is a rational function in $z^{-1}$ of degree $(\frac{p-1}{2},\frac{p-1}{2})$ , concluding the proof. ∎

3D. Conclusion

Proof of Section 3A.

We can now employ the functional equation from Section 3C to derive the closed expression for $F(w)$ in Section 3A. Set

\displaystyle G_{p}(w):=\prod_{j=0}^{\infty}\frac{1-w^{(p+1)p^{j}}}{(1-w^{p^{j}})(1+w^{p^{j+1}})}\;\in\mathbb{Q}((w))^{\times}.

(3D.1)

\displaystyle G_{p}(w)=\frac{1-w^{p+1}}{(1-w)(1+w^{p})}G_{p}(w^{p})=\left(1+\frac{1}{d_{p}(w+w^{-1})}\right)G_{p}(w^{p}).

Thus setting $F(w)=G_{p}(w)F_{\ast}(w)$ , Section 3C can be rewritten as

\displaystyle F_{\ast}(w)=F_{\ast}(w^{p})+\frac{w(1-w^{p-1})}{(1-w)(1+w^{p})G_{p}(w)}.

So we get

\displaystyle F_{\ast}(w)=\sum_{k=0}^{\infty}\frac{w^{p^{k}}(1-w^{p^{k}(p-1)})}{(1-w^{p^{k}})(1+w^{p^{k+1}})G_{p}(w^{p^{k}})},

or equivalently

\displaystyle F(w)=\sum_{k=0}^{\infty}\frac{w^{p^{k}}(1-w^{p^{k}(p-1)})}{(1-w^{p^{k}})(1+w^{p^{k+1}})}\frac{G_{p}(w)}{G_{p}(w^{p^{k}})}.

The conclusion of Section 3A then follows from substituting (iterations of) Equation 3D.1. ∎

4. Asymptotics of the generating function

Recall that $p$ is a fixed prime. In order to understand the sequence $b_{n}$ from Equation 1B.1 better, we focus on the generating function $H(z)$ as $z$ approaches the radius of convergence $2$ , see Section 3, as usual in the theory of asymptotics of generating functions.

4A. Asymptotics

Concretely, we will focus on the behavior for $w\downarrow 1$ of $F(w)=H(w+w^{-1})$ , viewed as a smooth function

\displaystyle F\colon(0,1)\to\mathbb{R}.

This is well-defined, as $(0,1)$ lies in $\Omega\subset\mathcal{C}$ from Equation 3A.1. This can be seen directly from the formula $(x^{2}+(y-1)^{2}-2)(x^{2}+(y+1)^{2}-2)>0$ , or the plot Equation 3A.1. We will now prove one of our main results:

Theorem \theTheorem.

We have

\displaystyle F(w)=F_{0}(w)\cdot(1-w)^{-\log_{p}(\frac{p+1}{2})}\cdot\big{(}1+o(1)\big{)}.

Here $F_{0}(w)\colon(0,1)\to\mathbb{R}_{>0}$ is real analytic, $F_{0}(w^{p})=F_{0}(w)$ and bounded away from $0$ and $\infty$ .

Proof.

Recall from Section 3C that we have

\displaystyle F(w)=\frac{w(1-w^{p-1})}{(1-w)(1+w^{p})}+\left(1+\frac{w(1-w^{p-1})}{(1-w)(1+w^{p})}\right)\cdot F(w^{p}).

A calculation gives

\displaystyle(1-w)(1+w^{p})\cdot F(w)=w(1-w^{p-1})+(1-w^{p+1})\cdot F(w^{p}).

This is a $2$ -Mahler function of degree $p$ , cf. Equation 2B.1, meaning it is of the form

\displaystyle F(w)=r_{1}(w)+r_{2}(w)\cdot F(w^{p}).

Let $\lambda$ be a variable. After clearing denominators so that the $r_{i}(w)$ are polynomials, the so-called characteristic polynomial (as recalled e.g. in [BC17]) of a Mahler functional equation as in Equation 2B.2 is

\displaystyle r_{0}(w)\cdot\lambda^{s-1}-r_{1}(w)\cdot\lambda^{s-2}-r_{2}(w)\cdot\lambda^{s-3}-\cdots-r_{s}(w)\cdot\lambda^{0}\in\mathbb{C}[w,\lambda].

This in our example is

\displaystyle(1-w)(1+w^{p})\cdot\lambda-(1-w^{p+1})

which has a root at $\lambda(w)=\frac{w^{p+1}-1}{(w-1)(1+w^{p})}$ . At the relevant singularity we get

\displaystyle\lim_{w\uparrow 1}\lambda(w)=\frac{p+1}{2},

which is the eigenvalue of the Mahler function $F(w)$ . Now the classical theory of Mahler functions, see e.g. [BC17, Theorem 1], implies that the log with base the degree of the Mahler equation of this eigenvalue is minus the exponent of $(1-w)$ in the asymptotic expansion. The remaining parts of the theorem follow also from [BC17, Theorem 1]. ∎

In the reminder of this section we make Section 4A more explicit.

4B. The oscillating factor

We define the following function on $(0,1)$ , rescaling $F$ :

\displaystyle\mathbf{F}(w):=\big{(}\ln(w^{-1})\big{)}^{\log_{p}(\frac{p+1}{2})}F(w)\colon(0,1)\to\mathbb{R}_{>0}.

Lemma \theLemma.

As a function on $(0,1)$ we have:

\displaystyle\mathbf{F}(w^{p^{-r}})=\left(\tfrac{p+1}{2}\right)^{-r}\big{(}\ln(w^{-1})\big{)}^{\log_{p}(\frac{p+1}{2})}\sum_{k=-r}^{\infty}\frac{w^{p^{k}}(1-w^{p^{k}(p-1)})}{(1-w^{p^{k}})(1+w^{p^{k+1}})}\prod_{j=-r}^{k-1}\frac{1-w^{(p+1)p^{j}}}{(1-w^{p^{j}})(1+w^{p^{j+1}})}.

Proof.

Directly from Section 3A. ∎

We will now show that there exists a function $\mathbf{F}_{0}(w)\colon(0,1)\to\mathbb{R}$ with $\lim_{r\to\infty}\mathbf{F}(w^{p^{-r}})=\mathbf{F}_{0}(w)$ , defined pointwise, satisfying $\mathbf{F}_{0}(w^{p})=\mathbf{F}_{0}(w)$ . To this end, consider the product in $\mathbb{Z}[[w]]$ given by

\displaystyle\Pi(w):=\prod_{j=1}^{\infty}\frac{2}{p+1}\frac{1-w^{(p+1)p^{-j}}}{(1-w^{p^{-j}})(1+w^{p^{-j+1}})}=\prod_{j=1}^{\infty}\frac{2}{p+1}\frac{1+w^{p^{-j}}+w^{2p^{-j}}+\dots+w^{pp^{-j}}}{1+w^{p^{-j+1}}}.

Define the power series

\displaystyle\mathbf{F}_{0}(w):=\big{(}\ln(w^{-1})\big{)}^{\log_{p}(\frac{p+1}{2})}\sum_{k=-\infty}^{\infty}\frac{w^{p^{k}}(1-w^{p^{k}(p-1)})}{(1-w^{p^{k}})(1+w^{p^{k+1}})}\Pi(w^{p^{k}})\left(\frac{p+1}{2}\right)^{k}.

Proposition \theProposition.

For $0<w<1$ , we have

\displaystyle\lim_{r\to\infty}\mathbf{F}(w^{p^{-r}})=\mathbf{F}_{0}(w).

Moreover, $\mathbf{F}_{0}\colon(0,1)\to\mathbb{R}_{>0}$ is a real analytic and oscillatory term, $\mathbf{F}_{0}(w^{p})=\mathbf{F}_{0}(w)$ , and is bounded away from $0$ and $\infty$ .

Remark \theRemark.

The function $\mathbf{F}_{0}$ in Section 4B is a rescaling of $F_{0}$ from Section 4A, so describes the oscillation of $F_{0}$ .

Proof of Section 4B.

It is easy to see that $\mathbf{F}_{0}(w)$ is well-defined for $w\in(0,1)$ : Firstly, the factors in the expression $\Pi(w^{p^{k}})$ converge to $1$ rapidly, and this implies that $\Pi(w^{p^{k}})$ itself converges to some number in $(0,1)$ , and we can assume that $\Pi(w^{p^{k}})$ is equal to $1$ . Second, the left term in the sum goes to $(p-1)/2$ and takes values in $\big{(}1,(p-1)/2\big{)}$ for negative $k$ , so the negative part of the sum converges. Finally, for positive $k$ the summands go to $0$ rapidly and the sum also converges.

As in the previous paragraph, $\Pi(w^{p^{k}})$ converges to $1$ for $w\in(0,1)$ . We then obtain

\displaystyle\mathbf{F}(w^{p^{-r}})\Pi(w^{p^{-r}})=\big{(}\ln(w^{-1})\big{)}^{\log_{p}(\frac{p+1}{2})}\sum_{k=-r}^{\infty}\frac{w^{p^{k}}(1-w^{p^{k}(p-1)})}{(1-w^{p^{k}})(1+w^{p^{k+1}})}\Pi(w^{p^{k}})\left(\frac{p+1}{2}\right)^{k}.

The claim follows. ∎

Proposition \theProposition.

The series $\mathbf{F}_{0}(p^{-v})$ converges absolutely and uniformly on compact sets in the region $\mathrm{Re}\,v>0$ but has a dense set of singularities on the imaginary axis.

Proof.

This follows as for $p=2$ proven in Section 4C below. Details are omitted. ∎

4C. Example for $p=2$

For $p=2$ , the expression for $\mathbf{F}_{0}$ simplifies to

\displaystyle\mathbf{F}_{0}(w)=\big{(}\ln(w^{-1})\big{)}^{\log_{2}(\frac{3}{2})}\sum_{k=-\infty}^{\infty}\left(\frac{3}{2}\right)^{k}\frac{1}{w^{2^{k}}+w^{-2^{k}}}\prod_{j=1}^{\infty}\frac{2}{3}\left(1+\frac{1}{w^{2^{k-j}}+w^{-2^{k-j}}}\right).

In particular, we have

\displaystyle\mathbf{F}_{0}(2^{-v})=\ln 2\sum_{k=-\infty}^{\infty}\frac{(2^{k}v)^{\log_{2}(\frac{3}{2})}}{2^{2^{k}v}+2^{-2^{k}v}}\Phi(2^{k}v),\text{ where }\Phi(v):=\prod_{j=1}^{\infty}\left(1-\frac{(2^{2^{-j-1}v}-2^{-2^{-j-1}v})^{2}}{3(2^{2^{-j}v}+2^{-2^{-j}v})}\right).

Lemma \theLemma.

The product $\Phi(v)$ converges absolutely and uniformly on compact sets (not containing zeros and poles) to a meromorphic function of $v\in\mathbb{C}$ .

Proof.

This holds since $2^{2^{-j}v}\to 1$ exponentially fast as $j\to\infty$ . ∎

The poles of factors in $\Phi$ are solutions of the equation $e^{2^{-j}v\ln 2}=\pm i$ , i.e.,

\displaystyle v=\frac{2^{j-1}(2n+1)\pi i}{\ln 2}.

On the other hand, zeros occur when $2^{2^{-j+1}v}$ nontrivial cube roots of $1$ , so

\displaystyle v=\frac{2^{j}(3n\pm 1)\pi i}{3\ln 2},

and they have multiplicities (the number of factors $2$ in $3n\pm 1$ ).

Proposition \theProposition.

The series $\mathbf{F}_{0}(2^{-v})$ converges absolutely and uniformly on compact sets in the region $\mathrm{Re}\,v>0$ but has a dense set of singularities on the imaginary axis.

Proof.

This holds by the above discussion. ∎

5. Monotonicity

In this section we address the monotonicity of our main sequence.

5A. Neighboring values of $b_{n}$

We will now prove:

Theorem \theTheorem.

We have $b_{n+2}\leq 4b_{n}$ , or equivalently $b_{n+2}/2^{n+2}\leq b_{n}/2^{n}$ , for all $n\in\mathbb{Z}_{\geq 0}$ .

Proof.

The proof will occupy this section, and is split into a few lemmas. For the first lemma up next, let $L_{n}(w)\in\mathbb{Z}[w,w^{-1}]$ for $n\in\mathbb{Z}_{\geq 0}$ be a sequence of Laurent polynomials such that, for some $\nu\in\mathbb{Z}$ , we have

(5A.1)

\displaystyle L_{n}(w^{-1})=w^{\nu}L_{n}(w)\quad\text{and}\quad L_{n+1}(w)+L_{n-1}(w)=(w+w^{-1})L_{n}(w),n\in\mathbb{Z}_{\geq 1}.

Note that $L_{n}/L_{n+1}\in\mathbb{Q}(z)\subset\mathbb{Q}[[z^{-1}]]$ , where we use $\mathbb{Q}(z)\subset\mathbb{Q}(w)$ as in Section 3A. Now we will prove that actually $L_{n}/L_{n+1}\in\mathbb{Z}_{\geq 0}[[z^{-1}]]$ .

Lemma \theLemma.

If $L_{0}/L_{1}$ is a positive power series in $z^{-1}$ , then the same is true for $L_{n}/L_{n+1}$ for all $n\in\mathbb{Z}_{\geq 0}$ .

Proof.

We have

\displaystyle\frac{L_{n}}{L_{n+1}}=\frac{L_{n}}{zL_{n}-L_{n-1}}=\frac{z^{-1}}{1-z^{-1}\frac{L_{n-1}}{L_{n}}}.

So the statement follows by induction on $n$ . ∎

For any $n\in\frac{1}{2}\mathbb{Z}_{>0}$ and an integer $1\leq r\leq n$ , let us define

\displaystyle K_{r,n}(z):=\frac{w^{n-r}+w^{-n+r}}{w^{n}+w^{-n}}\in\mathbb{Z}[[z^{-1}]].

Similarly, define

\displaystyle M_{r,n}(z):=\frac{w^{n-r}+w^{n-r-2}+...+w^{-n+r}}{w^{n}+w^{n-2}+...+w^{-n}}=\frac{w^{n-r+1}-w^{-n+r-1}}{w^{n+1}-w^{-n-1}}\in\mathbb{Z}[[z^{-1}]].

Lemma \theLemma.

(a)

The function $K_{r,n}(z)$ has positive Taylor coefficients in $z^{-1}$ .
(b)

The function $M_{r,n}(z)$ has positive Taylor coefficients in $z^{-1}$ .

Proof.

(a). If $n$ is an integer, let $L_{n}(w)=w^{n}+w^{-n}$ , and if $n$ is an honest half integer, let $L_{n-\frac{1}{2}}(w)=w^{\frac{1}{2}}(w^{n}+w^{-n})$ . Then $L_{n}$ satisfy (5A.1) with $\nu=0$ or $\nu=1$ . Moreover, $L_{0}/L_{1}=2z^{-1}$ in the integer case and $L_{0}/L_{1}=\frac{1}{z-1}=\frac{z^{-1}}{1-z^{-1}}$ in the non-integer case.

Thus, the result follows from Section 5A, as we can write

\displaystyle K_{r,n}(z)=\frac{L_{n-r}(z)}{L_{n}(z)}=\prod_{j=1}^{r}\frac{L_{n-j}(z)}{L_{n-j+1}(z)}.

(b). Let $L_{n}(w)=w^{n}+w^{n-2}+\dots+w^{-n}$ . Then $L_{n}$ satisfy (5A.1), and $L_{0}/L_{1}=z^{-1}$ . Thus, as in part (a), the result follows from Section 5A. ∎

We let

\displaystyle f(z)=\sum_{n\geq 0}\frac{b_{n}}{2^{n}}z^{-n-1}=2H(2z).

Moreover, we consider the renormalization

\displaystyle\psi(z):=-(1-z^{-2})f(z)=\sum_{n\geq 0}c_{n}z^{-n-1}.

Note that $b_{0}=b_{1}=1$ , so that $c_{0}=-1$ and $c_{1}=-1/2$ are not positive. But we prove:

Lemma \theLemma.

The Taylor coefficients $c_{n}$ in $z^{-1}$ of the function $\psi(z)$ satisfy $c_{n}>0$ for $n>1$ .

Example \theExample.

For $p=2$ the function $\psi(z)$ Taylor expands in $z^{-1}$ as

\displaystyle\psi(z)=-z-\frac{1}{2}z^{2}+\frac{3}{4}z^{3}+\frac{1}{8}z^{4}+\frac{1}{16}z^{5}+\frac{3}{32}z^{6}+\frac{3}{64}z^{7}+\frac{7}{128}z^{8}+\frac{7}{256}z^{9}+\dots

which can be obtained from the data listed in Section 9A.

Proof of Section 5A.

Recall the 2-Mahler coefficients $d_{p}(z)$ from Section 3C, and also the functional equation for $H(z)$ given in the proof of Section 3C, namely:

\displaystyle H(z)=\frac{1}{d_{p}(z)}+\left(1+\frac{1}{d_{p}(z)}\right)\cdot H\big{(}2T_{p}(z/2)\big{)}.

Here $T_{m}(x)$ is the $m$ th Chebyshev polynomial of the first kind with $T_{-1}=0$ , as recalled in Equation 3B.3. Set also $s_{p}(z)=d_{p}(2z)$ .

Generalzing the calculation in Section 5B below, we get the following expression (note that $\psi(z)=-2(1-z^{-2})H(2z)$ which gives a rescaling of the above functional equation):

\displaystyle\psi(z)=-2\frac{1-z^{-2}}{s_{p}(z)}+\left(1+\frac{1}{s_{p}(z)}\right)\frac{1-z^{-2}}{1-T_{p}(z)^{-2}}\cdot\psi\big{(}T_{p}(z)\big{)}.

We let $M_{n}(z)=2\big{(}T_{n}(z)+T_{n-2}(z)+\dots+T_{0\text{ or 1}}(z)\big{)}$ , with the last terms being either $T_{0}(z)$ or $T_{1}(z)$ , depending on the parity. A calculation gives

\displaystyle\frac{1-z^{-2}}{1-T_{p}(z)^{-2}}=\frac{z^{-2}T_{p}(z)^{2}}{M_{p-1}(z)^{2}}.

To see this observe that this is equivalent to $\big{(}T_{p}(z)^{2}-1\big{)}/(z^{2}-1)=M_{p-1}(z)^{2}$ . This in turn follows by setting $2z=w+w^{-1}$ and we get

	$\displaystyle\frac{T_{p}(z)^{2}-1}{z^{2}-1}=\frac{(w^{p}+w^{-p})^{2}-4}{(w+w^{-1})^{2}-4}=\left(\frac{w^{p}-w^{-p}}{w-w^{-1}}\right)^{2}$
	$\displaystyle=\left(w^{p-1}+w^{p-3}+\dots+w^{-p+3}+w^{-p+1}\right)^{2}=M_{p-1}(z)^{2}.$

We thus get

\displaystyle\psi(z)=-2\frac{1-z^{-2}}{s_{p}(z)}+\left(1+\frac{1}{s_{p}(z)}\right)\frac{z^{-2}T_{p}(z)^{2}}{M_{p-1}(z)^{2}}\cdot\psi\big{(}T_{p}(z)\big{)}.

We get:

	$\displaystyle-z^{-1}-\frac{1}{2}z^{-2}+\psi_{\geq 2}(z)=$	$\displaystyle-2\frac{1-z^{-2}}{s_{p}(z)}-\left(1+\frac{1}{s_{p}(z)}\right)\frac{z^{-2}\big{(}2T_{p}(z)+1\big{)}}{4M_{p-1}(z)^{2}}$
		$\displaystyle+\left(1+\frac{1}{s_{p}(z)}\right)\frac{z^{-2}T_{p}(z)^{2}}{M_{p-1}(z)^{2}}\cdot\psi_{\geq 2}\big{(}T_{p}(z)\big{)}.$

We rewrite this as

\displaystyle\psi_{\geq 2}(z)=R(z)+S(z)\frac{z^{-2}T_{p}(z)^{2}}{M_{p-1}(z)^{2}}\cdot\psi_{\geq 2}\big{(}T_{p}(z)\big{)},

where

	$\displaystyle R(z)=z^{-1}+\frac{1}{2}z^{-2}-2\frac{1-z^{-2}}{s_{p}(z)}-\left(1+\frac{1}{s_{p}(z)}\right)\frac{z^{-2}\big{(}2T_{p}(z)+1\big{)}}{4M_{p-1}(z)^{2}},$
	$\displaystyle S(z)=1+\frac{1}{s_{p}(z)}.$

Recall that the polynomial $T_{m}(z)$ has real roots. Furthermore, $T_{m}(z)$ or $T_{m}(z)/z$ depends on $z^{2}$ . Both observations together imply that we have that $1/T_{m}(z)$ is a positive series in $z^{-1}$ . The same applies to $M_{m}(z)$ . So it suffices to check that the series $S(z)$ and $R(z)$ are positive. The next two lemmas imply these facts.

Lemma \theLemma.

The function $d_{p}(z)^{-1}=\frac{w(1-w^{p-1})}{(1-w)(1+w^{p})}$ has positive Taylor coefficients in $z^{-1}$ . The same holds for $s_{p}(z)^{-1}$ .

Proof.

For $p=2$ we have

\displaystyle\frac{1}{d_{p}(z)}=\frac{w(1-w)}{(1-w)(1+w^{2})}=\frac{w}{1+w^{2}}=\frac{1}{z},

which is manifestly positive.

Hence, we can assume that $p$ is odd. We have

\displaystyle\frac{1}{d_{p}(z)}=\frac{w+w^{2}+\dots+w^{p-1}}{1+w^{p}}=K_{1,\frac{p}{2}}+K_{2,\frac{p}{2}}+\dots+K_{\frac{p-1}{2},\frac{p}{2}}.

So the result follows from Section 5A.(a).

That $s_{p}(z)^{-1}$ is positive is then immediate from $s_{p}(z)=d_{p}(2z)$ . ∎

Lemma \theLemma.

The function $z^{2}R(z)$ has positive Taylor coefficients in $z^{-1}$ .

Proof.

A calculation shows that

	$\displaystyle z^{2}R(z)=\underbrace{2\frac{w(1-w^{2p-2})}{1-w^{2p}}}_{Y_{1}}+\underbrace{\frac{w^{2}(1-w^{2p-4})}{1-w^{2p}}}_{Y_{2}}+\underbrace{\frac{w^{p-1}(1-w^{2})}{1-w^{2p}}}_{Y_{3}}$
	$\displaystyle+\underbrace{\frac{w^{p}(1-w^{2p-2})(1-w^{2})}{(1-w^{2p})^{2}}}_{Y_{4}}+\underbrace{\frac{w(1-w^{p-1})}{(1-w)(1+w^{p})}\frac{w^{2p-2}(1-w^{2})^{2}}{(1-w^{2p})^{2}}}_{Y_{5}}.$

The $Y_{j}$ , as indicated above, are positive for $j=1,2,3,4,5$ . Indeed, $Y_{1}$ , $Y_{2}$ , $Y_{3}$ , $Y_{4}$ are positive by Section 5A and $Y_{5}$ is positive by Section 5A and Section 5A. ∎

Section 5A implies that $R(z)$ is positive, while $S(z)$ is positive by Section 5A. This finishes the proof of Section 5A. ∎

Taking all together yields Section 5A. ∎

5B. Example for $p=2$

For $p=2$ the calculation in Section 5A is rather straightforward. In this case Section 3C implies the functional equation

	$\displaystyle\psi(z)$	$\displaystyle=-(1-z^{-2})f(z)=-2(1-z^{-2})H(2z)$
		$\displaystyle=-2(1-z^{-2})\big{(}\tfrac{1}{2z}+(1+\tfrac{1}{2z})\cdot H(2T_{2}(z))\big{)}$
		$\displaystyle=-2(1-z^{-2})\big{(}\tfrac{1}{2z}+\tfrac{1}{2}(1+\tfrac{1}{2z})\tfrac{(2z^{2}-1)^{2}}{4z^{4}}\cdot f(2z^{2}-1)\big{)}$
		$\displaystyle=-z^{-1}+z^{-3}+(1+\tfrac{1}{2z})\tfrac{(2z^{2}-1)^{2}}{4z^{4}}\cdot\psi(2z^{2}-1)$
		$\displaystyle=-2\frac{1-z^{-2}}{s_{2}(z)}+\left(1+\frac{1}{s_{2}(z)}\right)\frac{z^{-2}T_{2}(z)^{2}}{M_{1}(z)^{2}}\cdot\psi\big{(}T_{2}(z)\big{)}.$

This gives a recursion of the coefficients $c_{n}$ , namely:

\displaystyle\sum_{n\geq 2}c_{n}z^{-n-1}=\frac{3z^{-3}}{4}+\frac{z^{-4}}{8}+\frac{z^{-5}}{16}+\frac{1}{4}z^{-4}\left(1+\frac{1}{2}z^{-1}\right)\sum_{m\geq 2}c_{m}\frac{(2z^{2})^{-m+1}}{(1-\frac{1}{2z^{2}})^{m-1}}.

This recursion has positive coefficients, hence, Section 5A follows.

6. The main theorem

We now prove Section 1B after an auxiliary lemma for which we use Equation 1B.3.

6A. A Tauberian lemma

The following type of result, often called Tauberian theory, is standard and just reformulated to suit our needs, see for instance [Tit58, §7.53] or [BGT89, Theorem 2.10.2] for related results.

Lemma \theLemma.

(a)

Consider a sequence $(a_{n})_{n\in\mathbb{Z}_{>0}}$ with $a_{n}\in\mathbb{R}_{\geq 0}$ , for which the series $f(z)=\sum_{n=1}^{\infty}a_{n}z^{n}$ has radius of convergence $1$ . If, for some $t\in(0,1)$ , we have,

$\displaystyle f(z)\;\asymp_{[t,1)}\;(1-z)^{-\beta},$

for some $\beta\in\mathbb{R}_{>0}$ , then:

$\displaystyle\sum_{k=1}^{n}a_{k}\;\asymp_{\mathbb{Z}_{>0}}\;n^{\beta}.$
(b)

Assume that, additionally to (a), for some $r\in\mathbb{Z}_{>0}$ and $B\in\mathbb{R}_{>0}$ we have

$\displaystyle a_{n+r}\leq a_{n}\leq B\cdot a_{n+r-1},$

for all $n\in\mathbb{Z}_{>0}$ . Then:

$\displaystyle a_{n}\;\asymp_{\mathbb{Z}_{>0}}\;n^{\beta-1}.$

Proof.

Claim (a) We set $s_{n}:=\sum_{k=1}^{n}a_{k}$ .

We first prove the upper bound on $s_{n}$ . For every $z\in[t,1)$ we have

\displaystyle s_{n}z^{n}=\sum_{k=1}^{n}a_{k}z^{n}\leq\sum_{k=1}^{n}a_{k}z^{k}\leq f(z)\leq\frac{A}{(1-z)^{\beta}},

for some $A\in\mathbb{R}_{>0}$ . The function $1/\big{(}z^{n}(1-z)^{\beta}\big{)}$ attains its minimum at $z=n/(\beta+n)$ (which is larger than $t$ for $n$ sufficiently large), allowing us to reformulate the above inequality as

\displaystyle s_{n}\leq A\frac{(\beta+n)^{\beta+n}}{\beta^{\beta}n^{n}}.

The latter then implies that for some $C>0$ , we have $s_{n}\leq C\cdot n^{\beta}$ .

Deriving the inequality in the other direction is more subtle. However, as explained in [Dus20], this (in fact, both inequalities) is a special case of the de Haan–Stadtmüller theorem, see [BGT89, Theorem 2.10.2].

Claim (b) By the conclusion of part (a), we know that for some $C>1$

\displaystyle C^{-1}\cdot n^{\beta}\leq\sum_{k=1}^{n}a_{k}\leq C\cdot n^{\beta},\quad\text{for all }n>0.

We will use this freely.

We again start with the upper bound. For all $n\geq r$ we find from monotonicity that

\displaystyle\tfrac{n}{r}a_{n}\leq\sum_{j=0}^{\lfloor(n-1)/r\rfloor}a_{n-rj}\leq C\cdot n^{\beta}.

From this we can derive $a_{n}\leq C^{\prime}\cdot n^{\beta-1}$ for all $n\in\mathbb{Z}_{>0}$ for $C^{\prime}=rC>1$ .

For the lower bound we consider the case $r=1$ first (in which case we can take $B=1$ and the second inequality in (b) is trivial). For any $N\geq n$ we have

\displaystyle(N-n)a_{n}\geq\sum_{i=n+1}^{N}a_{i}\geq C^{-1}\cdot N^{\beta}-C\cdot n^{\beta}.

Since $\beta>0$ , we can choose $m\in\mathbb{Z}_{>0}$ for which $m^{\beta}>C^{2}$ . For $N=mn$ , we can rewrite the above inequality as

\displaystyle a_{n}\geq\frac{C^{-1}m^{\beta}-C}{m-1}\cdot n^{\beta-1},\quad\text{for all }n>0.

By construction the factor in front of $n^{\beta-1}$ is positive and we are done.

In case $r>1$ we can similarly show that

(6A.1)

\displaystyle a_{n}+a_{n-1}+\cdots+a_{n-r+1}\geq A\cdot n^{\beta-1}

for some $A>0$ . On the other hand, we have

\displaystyle r\cdot a_{n}\geq a_{n}+B^{-1}\cdot a_{n+1-r}+\dots+B^{1-r}\cdot a_{n-(r-1)^{2}}\geq a_{n}+\sum_{i=1}^{r}B^{-i}\cdot a_{n+i-r}.

In particular, we can take $A^{\prime}>0$ so that

\displaystyle a_{n}\geq A^{\prime}\cdot(a_{n}+a_{n-1}+\dots+a_{n+1-r}),

which together with (6A.1) concludes the proof. ∎

Remark \theRemark.

The conclusion in Section 6A.(b) does not follow without the additional assumption $a_{n}\geq B\cdot a_{n+r-1}$ in case $r>1$ . Indeed, it suffices to take the sequence

\displaystyle a_{n}=\begin{cases}1&\text{if $n$ is odd},\\ 0&\text{if $n$ is even}.\end{cases}

Then $\sum_{k=0}^{n}a_{k}\asymp_{\mathbb{Z}_{\geq 0}}n$ and $a_{n+2}\leq a_{n}$ , but $a_{n}\not\asymp_{\mathbb{Z}_{\geq 0}}1$ .

We in fact already had an example of this type: in Section 2BI the Cantor set sequence $(\mathrm{ca}_{n})_{n\in\mathbb{Z}_{\geq 0}}$ satisfies

\displaystyle\sum_{n\in\mathbb{Z}_{\geq 0}}\mathrm{ca}_{n}w^{n}\asymp_{w\uparrow 1}(1-w)^{-\tau}\big{(}1+o(1)\big{)}\quad\text{and}\quad\sum_{k=0}^{n}\mathrm{ca}_{k}\asymp n^{\tau}\big{(}1+o(1)\big{)},

for $\tau=\log_{3}2$ . However, $\mathrm{ca}_{n}\not\asymp_{\mathbb{Z}_{\geq 0}}n^{\tau-1}$ .

6B. Conclusion

We are ready to prove our main result:

Proof of Section 1B.

Set $a_{n}:=\frac{b_{n-1}}{2^{n-1}}$ for $n\in\mathbb{Z}_{>0}$ . Then Section 5A implies that $a_{n+2}\leq a_{n}$ . We also know that $b_{n}\leq b_{n+1}$ , since each indecomposable summand in $V^{\otimes n}$ is responsible for at least one in $V^{\otimes n+1}$ . Hence, $a_{n}\leq 2\cdot a_{n+1}$ , so the sequence $a_{n}$ satisfies all conditions in parts (a) and (b) of Section 6A with $r=2$ .

For the purpose of this proof we will write $\asymp$ to mean $\asymp_{[t,1)}$ for an arbitrary $t\in(0,1)$ . Recall that

\displaystyle F(w)=\sum_{n\geq 0}b_{n}(w+w^{-1})^{-n-1}=H(w+w^{-1}).

Section 4A implies that

\displaystyle F(w)\asymp(1-w)^{-2\beta},\quad\beta=\frac{1}{2}\log_{p}\left(\frac{p+1}{2}\right)=t_{p}+1.

Recall further that

\displaystyle H(z)=\sum_{n\geq 0}b_{n}z^{-n-1}\quad\text{and thus}\quad 2H(2z^{-1})=\sum_{n>0}a_{n}z^{n}.

Solving $2z^{-1}=w+w^{-1}$ for $w$ in the region $w<1$ then yields

\displaystyle 2H(2z^{-1})\asymp\bigg{(}\frac{\sqrt{1-z^{2}}+z-1}{z}\bigg{)}^{-2\beta}=(1-z)^{-\beta}\bigg{(}\frac{z^{2}}{2(1-\sqrt{1-z^{2}})}\bigg{)}^{\beta}.

Since the last factor is bounded on $(0,1]$ away from $0$ , we find

\displaystyle\sum_{n>0}a_{n}z^{n}=2H(\tfrac{2}{z})\asymp(1-z)^{-\beta}.

Section 6A.(b) then implies

\displaystyle a_{n}\asymp_{\mathbb{Z}_{>0}}n^{\beta-1}=n^{t_{p}},

and we are done. ∎

7. Additional results in characteristic two

Throughout this section let $p=2$ . This restriction is mostly for convenience: with some work the statements and proofs below generalize to all primes.

7A. Neighboring values of $b_{n}$ in characteristic two

We will now strengthen Section 5A, where we can focus on difference two for the indexes since $b_{2n-1}=b_{2n}$ for $n\in\mathbb{Z}_{>0}$ . The auxiliary sequence that we use is $d_{n}=1-\frac{b_{n+2}}{4b_{n}}$ , for $n\in\mathbb{Z}_{\geq 0}$ .

Proposition \theProposition.

We have $d_{n}\geq 0$ and $d_{n}\in O(n^{-t_{2}-1})$ .

Since $-t_{2}-1\approx-0.293$ , Section 7A shows that $\lim_{n\to\infty}d_{n}=0$ .

Proof of Section 7A.

We start with an analysis of $c_{n}:=b_{n+4}-8b_{n+2}+16b_{n}$ .

Lemma \theLemma.

We have $0\leq c_{n}$ for all $n\in\mathbb{Z}_{\geq 0}$ .

Proof.

By Section 3C, we have $H(z)=\sum_{n\geq 0}\frac{b_{n}}{2^{n}}z^{-n-1}=z^{-1}+(1+z^{-1})H(z^{2}-2)$ . So, if $\eta(z)=(1-4z^{-2})^{2}H(z)=z^{-1}+z^{-2}-7z^{-3}-5z^{-4}+\xi(z)$ , then we get

	$\displaystyle z^{-1}+z^{-2}-7z^{-3}-5z^{-4}+\xi(z)=z^{-1}(1-4z^{-2})^{2}$
	$\displaystyle+z^{-8}(1+z^{-1})\big{(}(z^{2}-2)^{3}+(z^{2}-2)^{2}-7(z^{2}-2)-5+(z^{2}-2)^{4}\xi(z^{2}-2)\big{)}.$

Hence, we get for $\xi(z)=\sum_{n=0}^{\infty}c_{n}z^{-n-5}$ that:

\displaystyle\sum_{n=0}^{\infty}c_{n}z^{-n-5}=11z^{-5}+z^{-6}+z^{-7}+5z^{-8}+5z^{-9}+(1+z^{-1})\sum_{n=0}^{\infty}c_{n}z^{-2(n+5)}(1-2z^{-2})^{-n-1}.

This gives a positive recursion for $c_{n}$ , which implies the statement. ∎

The numbers $d_{n}$ are nonnegative by Section 5A. Moreover, Section 7A implies that $2\leq 1-d_{n+2}+\frac{1}{1-d_{n}}$ which gives

\displaystyle 1-d_{n}\leq\frac{1}{1+d_{n+2}}.

This yields $1-d_{n-4}\leq\frac{1}{1+d_{n-2}}$ and $1-\frac{1}{1+d_{n}}\leq d_{n-2}$ . Putting these together gives $1-d_{n-4}\leq\frac{1}{1+1-\frac{1}{1+d_{n}}}=\frac{1+d_{n}}{1+2d_{n}}$ , and iterating this procedure then gives

\displaystyle 1-d_{n-2k}\leq\frac{1+(k-1)d_{n}}{1+kd_{n}}.

We also have, say for the even values,

\displaystyle(1-d_{0})\cdot\ldots\cdot(1-d_{n-2})=\tfrac{1}{4}\cdot a_{n}\geq\tfrac{1}{4}C_{1}\cdot n^{t_{2}}

for some $C_{1}\in\mathbb{R}_{>0}$ , where the final inequality is Section 1B. Thus,

\displaystyle C_{1}^{\prime}\cdot n^{t_{2}}\leq\frac{1}{1+nd_{n}},

for some $C_{1}^{\prime}\in\mathbb{R}_{>0}$ , which, by rewriting, proves the statement. ∎

7B. The main theorem revisited

The following generalizes Section 1B:

Proposition \theProposition.

Equation 1D.1 is true.

Proof.

Let $a_{n}^{\prime}=\frac{b_{2n}}{2^{2n}}$ . Note that this ignores the odd values since $a_{n}$ runs only over the even values of $b_{n}$ , but since $b_{2n-1}=b_{2n}$ for all $n\in\mathbb{Z}_{>0}$ this does not play a role below and just simplifies the notation.

We observe that $4^{t_{2}}=4^{-\frac{1}{2}\log_{2}\frac{8}{3}}=\frac{3}{8}$ , so that Section 1B implies that we can sandwich both, $\tfrac{3}{8}\cdot a_{n}^{\prime}$ and $a_{4n}^{\prime}$ , at the same time:

\displaystyle C_{1}\cdot\tfrac{3}{8}(2n)^{t_{2}}\leq\left\{\begin{gathered}\tfrac{3}{8}\cdot a_{n}^{\prime}\\ a_{4n}^{\prime}\end{gathered}\right.\leq C_{2}\cdot\tfrac{3}{8}(2n)^{t_{2}}.

However, this does not imply the result yet, we need to know a bit more about the sequence $a_{n}^{\prime}$ . Firstly, it follows from Section 7A that there is a constant $C\in\mathbb{R}_{>0}$ such that

(7B.1)

\displaystyle(1-\tfrac{C}{n})a_{n}^{\prime}<a_{n+1}^{\prime}<a_{n}^{\prime}.

Also $a_{0}=1$ and, by Section 3B, for $n>0$ we have

\displaystyle a_{n}^{\prime}=2^{-n-1}\left(\sum_{k=0}^{\lfloor(n-1)/2\rfloor}\bigg{(}\binom{n-1}{2k}+2\binom{n-1}{2k-1}\bigg{)}a_{k}^{\prime}+\delta_{0,n\bmod 2}\cdot 2a_{n/2}^{\prime}\right).

Thus, when running over even values, we get

(7B.2)

\displaystyle a_{2n}^{\prime}=2^{-2n-1}\sum_{k=0}^{n}\bigg{(}\binom{2n-1}{2k}+2\binom{2n-1}{2k-1}\bigg{)}a_{k}^{\prime}.

Note also that, for $n>0$ , we have

\displaystyle 2^{-2n-1}\sum_{k=0}^{n}\bigg{(}\binom{2n-1}{2k}+2\binom{2n-1}{2k-1}\bigg{)}=\frac{3}{8}.

So using this, combined with (7B.1) and (7B.2), we get

\displaystyle a_{4n}^{\prime}\sim\tfrac{3}{8}\cdot a_{n}^{\prime}.

Hence, for all $x\in\mathbb{R}_{>0}$ there exists a limit

\displaystyle\lim_{m\to\infty}a_{\lfloor 4^{m}x\rfloor}^{\prime}(4^{m}x)^{-t_{2}}=\mathbf{f}_{0}(x),

and $\mathbf{f}_{0}$ is continuous on $\mathbb{R}_{>0}$ with $\mathbf{f}_{0}(4x)=\mathbf{f}_{0}(x)$ , and bounded away from $0$ and $\infty$ . ∎

8. Fourier coefficients

The functions that we have met above are oscillating, and in this section we analyze their Fourier coefficients.

8A. An Abelian lemma

Tauberian theory as in Section 6A, roughly speaking, says that given a certain behavior of a generating function, we get a certain behavior of the associated sequence. There is an “inverse” to Tauberian theory often called Abelian theory.

Below we will need the following well-known result from Abelian theory:

Lemma \theLemma.

Assume that we have two functions $F(q)=\sum_{n=1}^{\infty}a_{n}q^{n}\in\mathbb{R}_{\geq}[[q]]$ and $G(q)=\sum_{n=1}^{\infty}b_{n}q^{n}\in\mathbb{R}_{\geq}[[q]]$ which converge for $q\in[0,1)$ and diverge for $q=1$ . Then

\displaystyle\big{(}a_{n}\sim_{n\to\infty}b_{n}\big{)}\Rightarrow\big{(}F(q)\sim_{q\uparrow 1}G(q)\big{)}.

Proof.

This is for example explained at the beginning of [Tit58, Section 7.5]. ∎

8B. Some generalities on asymptotics of Fourier coefficients

Let $\nu\in\mathbb{Z}_{\geq 2}$ and $\bar{t}>-1$ . Suppose we have a sequence $(a_{n})_{n\in\mathbb{Z}_{>0}}$ such that we have asymptotically

\displaystyle a_{n}\sim\tilde{h}(\log_{\nu}n)\cdot n^{\bar{t}}

for some continuous $1$ -periodic function $\tilde{h}\colon\mathbb{R}\to\mathbb{R}_{>0}$ with $\tilde{h}(x+1)=\tilde{h}(x)$ . The associated generating function is the series

\displaystyle f(q)=\sum_{n=1}^{\infty}a_{n}q^{n}\in\mathbb{R}[[q]].

Lemma \theLemma.

The series $f(q)$ absolutely converges for $0\leq|q|<1$ with singularity at $q=1$ .

Proof.

This follows since $a_{n}\sim\tilde{h}(\log_{\nu}n)\cdot n^{\bar{t}}$ . ∎

As before, let $\Gamma$ denote the gamma function. Recall that the Fourier coefficient formula of a $P$ -periodic function $g$ (so that the integral expression up next makes sense) are given by $c_{n}=\frac{1}{P}\int_{-P/2}^{P/2}g(x)e^{-2\pi ixn/P}dx$ , for $n\in\mathbb{Z}$ . That is, $g(x)=\sum_{n=-\infty}^{\infty}c_{n}e^{2\pi ixn/P}$ for $x\in[-P/2,P/2]$ .

We get the following asymptotic of $f$ :

Proposition \theProposition.

Retain the assumptions above, and denote the Fourier coefficients of $h$ by $h_{n}$ . We have

\displaystyle f(q)\sim_{q\uparrow 1}\big{(}\ln(q^{-1})\big{)}^{-\bar{t}-1}L\big{(}\log_{\nu}(\ln(q^{-1}))\big{)}

where $L$ is the $1$ -periodic function given by

\displaystyle L(y)=\sum_{n\in\mathbb{Z}}\Gamma(\bar{t}+1-\tfrac{2\pi in}{\log\nu})e^{2\pi iny}h_{-n}.

Proof.

For the argument below we note that $q=1$ is the relevant singularity of $f$ , and we need to analyze the growth rate of $f$ for $q\uparrow 1$ .

Recalling that $a_{n}\sim\tilde{h}(\log_{\nu}n)\cdot n^{\bar{t}}$ , we use Section 8A and get

\displaystyle f(q)\sim_{q\uparrow 1}\sum_{n=1}^{\infty}\tilde{h}(\log_{\nu}n)n^{\bar{t}}q^{n}.

We rewrite this as

\displaystyle f(q)\sim_{q\uparrow 1}\sum_{r\in\mathbb{Z}_{\geq 0}}\sum_{1\leq\nu^{-r}n<\nu}\tilde{h}(\log_{\nu}n)n^{\bar{t}}q^{n}.

So setting $x=\nu^{-r}n$ and using the periodicity of $\tilde{h}$ , namely $\tilde{h}(x+1)=\tilde{h}(x)$ , we get

\displaystyle f(q)\sim_{q\uparrow 1}\sum_{r\in\mathbb{Z}_{\geq 0}}\nu^{r\cdot\bar{t}}\sum_{1\leq x<\nu}\tilde{h}(\log_{\nu}x)x^{\bar{t}}q^{\nu^{r}x},

where the inner sum is over (not necessarily reduced) fractions with denominator $\nu^{r}$ . Note that the main contribution to the asymptotics $q\uparrow 1$ comes from large $r$ , as $\bar{t}>-1$ . Hence, we may replace the inner sum by an integral times $\nu^{r}$ (reciprocal length of step), and the range $\mathbb{Z}_{\geq 0}$ of summation for the outer sum by $\mathbb{Z}$ . After doing this we get:

\displaystyle f(q)\sim_{q\uparrow 1}\sum_{r\in\mathbb{Z}_{\geq 0}}\nu^{r(\bar{t}+1)}\int_{1}^{\nu}\tilde{h}(\log_{\nu}x)x^{\bar{t}}q^{\nu^{r}x}dx.

Making the substitution $u=\log_{\nu}(x)$ allows us to rewrite the above as

\ln(\nu)\sum_{r\in\mathbb{Z}_{\geq 0}}\int_{0}^{1}\nu^{(u+r)(\bar{t}+1)}q^{\nu^{r+u}}\tilde{h}(u)du\;=\;\ln(\nu)\int^{\infty}_{0}\nu^{u(\bar{t}+1)}q^{\nu^{u}}\tilde{h}(u)du.

Substituting back to $x$ , the above yields

\displaystyle f(q)\sim_{q\uparrow 1}\int_{0}^{\infty}x^{\bar{t}}q^{x}\tilde{h}(\log_{\nu}(x))dx.

Replacing the dummy variable $x$ by $x/\ln(q^{-1})$ yields

\displaystyle f(q)\sim_{q\uparrow 1}(\ln(q^{-1}))^{-\bar{t}-1}\int_{0}^{\infty}x^{\bar{t}}e^{-x}\tilde{h}\left(\log_{\nu}(x)-\log_{\nu}(\ln(q^{-1}))\right)dx.

We can thus define the 1-periodic function

L(y)\;:=\;\int_{0}^{\infty}x^{\bar{t}}e^{-x}\tilde{h}\left(\log_{\nu}(x)-y\right)dx.

That this periodic function corresponds to the one in the proposition follows easily from the definition $\Gamma(z)=\int_{0}^{\infty}x^{z-1}e^{-x}dx$ of the Gamma function. ∎

Theorem \theTheorem.

With the notation as in Section 8B, we have

\displaystyle h_{n}=\frac{L_{n}}{\Gamma(\bar{t}+1+\frac{2\pi in}{\log\nu})}

for the Fourier coefficients $L_{n}$ of $L(y)$ .

Proof.

By Section 8B and Fourier inversion. ∎

8C. Analysis of the Fourier coefficients of the generating function for $b_{n}/2^{n}$

Assume that Equation 1D.1 holds. (For $p=2$ this assumption is true by Section 7.) We now apply the results in Section 8B to our example of $b_{n}/2^{n}$ in characteristic $p$ . That is, we look at $a_{n}=b_{n}/2^{n}$ for the $a_{n}$ in Section 8B.

Consider the generating function of $a_{n}$ :

\displaystyle f(q)=\sum_{n=0}^{\infty}a_{n}q^{n}\in\mathbb{R}_{>0}[[q]].

The relevant singularity is at $q=1$ . Below we will also use the generating function in $w$ defined mutatis mutandis as in Section 3A.

Moreover, associated to $f$ we can define $\mathbf{f}_{0}$ similarly as $\mathbf{F}_{0}$ from Section 4B but defined now in $q$ , and we get the same type of statements for $\mathbf{f}_{0}$ as for $\mathbf{F}_{0}$ .

Recall $t_{p}=\frac{1}{2}\log_{p}\frac{2p^{2}}{p-1}$ from Equation 1B.2. Setting $\nu=2p$ (which we expect to be correct based on numerical data) and $\bar{t}=t_{p}$ , we get:

Proposition \theProposition.

We have

\displaystyle f\sim_{q\uparrow 1}\big{(}\ln(q^{-1})\big{)}^{-t_{p}-1}\mathbf{f}_{0}\big{(}\log_{2p}\ln(q^{-1})\big{)}

and the Fourier coefficients of $\mathbf{f}_{0}$ are given by Section 9C.

Proof.

Recall from Section 4B that $\lim_{r\to\infty}(\ln w^{-p^{-r}})^{\log_{p}(\frac{p+1}{2})}F(w^{p^{-r}})=\mathbf{F}_{0}(w)$ . We can write this as $\lim_{q\uparrow 1}\big{(}\ln(q^{-1})\big{)}^{-2(t_{p}+1)}f(q)=\mathbf{f}_{0}\big{(}\log_{2p}\ln(q^{-1})\big{)}$ . Then Section 8B as well as Section 8B apply. ∎

9. Numerical data

All of the below can be found on the GitHub page associated to this project [CEOT24], where the reader can find code that can be run online. For convenience, we list a few numerical examples here as well.

Remark \theRemark.

All plots below are logplots (also called semi-log plots). This means that the plots have one axis (the y-axis for us) on a logarithmic scale, the other on a linear scale.

9A. The main sequence

Below we will always use $p\in\{2,3,5,7,\infty\}$ with $p=\infty$ meaning that the characteristic is large compared to the considered range, which leads to the same data as in characteristic zero.

We first give tables for $b_{n}$ for $n\in\{0,\dots,15\}$ :

	$b_{0}$	$b_{1}$	$b_{2}$	$b_{3}$	$b_{4}$	$b_{5}$	$b_{6}$	$b_{7}$	$b_{8}$	$b_{9}$	$b_{10}$	$b_{11}$	$b_{12}$	$b_{13}$	$b_{14}$	$b_{15}$
$p=2$	1	1	1	3	3	9	9	29	29	99	99	351	351	1273	1273	4679
$p=3$	1	1	2	2	5	6	15	21	50	77	176	286	637	1066	2340	3978
$p=5$	1	1	2	3	6	9	19	28	62	91	208	308	716	1079	2522	3886
$p=7$	1	1	2	3	6	10	20	34	69	117	242	407	858	1431	3069	5085
$p=\infty$	1	1	2	3	6	10	20	35	70	126	252	462	924	1716	3432	6435

For completeness, we give here the $b_{n}$ for $n\in\{0,\dots,30\}$ (in order top to bottom, excluding $p=\infty$ where the sequence is [OEI23, A001405]) in a copy-able fashion:

$\{1,1,1,3,3,9,9,29,29,99,99,351,351,1273,1273,4679,4679,17341,17341,64637,64637,242019,242019,909789,909789,3432751,3432751,12998311,12998311,49387289,49387289\}$ .

$\{1,1,2,2,5,6,15,21,50,77,176,286,637,1066,2340,3978,8670,14859,32301,55575,120822,208221,453399,781794,1706301,2942460,6438551,11103665,24357506,42015664,92376280\}$ .

$\{1,1,2,3,6,9,19,28,62,91,208,308,716,1079,2522,3886,9061,14297,33098,53448,122551,202181,458757,771443,1732406,2962284,6587959,11428743,25193027,44250404,96775581\}$ .

$\{1,1,2,3,6,10,20,34,69,117,242,407,858,1431,3069,5085,11066,18258,40205,66215,147136,242420,542202,895390,2011165,3334125,7505955,12507121,28174255,47229893,106315770\}$ .

Here is one picture to compare their growth, including the case $p=\infty$ :

\displaystyle\leavevmode\hbox to233.59pt{\vbox to149.04pt{\pgfpicture\makeatletter\hbox{\hskip 116.7969pt\lower-74.51895pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-113.4639pt}{-71.18594pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=142.26378pt]{figs/ball}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.

Note that the sequences get a bit more regular if one runs over the, say, even values. This is in particular true in characteristic $p=2$ where $b_{2n-1}=b_{2n}$ . Below we will strategically sometimes only illustrates the even values.

Next, the logplots of $b_{n}/2^{n}$ compared with $n^{t_{p}}$ , and of $b_{n}$ compared with $n^{t_{p}}\cdot 2^{n}$ are:

	$\displaystyle\leavevmode\hbox to223.48pt{\vbox to137.47pt{\pgfpicture\makeatletter\hbox{\hskip 111.738pt\lower-68.73735pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-108.405pt}{-65.40434pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=130.88284pt]{figs/plott2}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-22.79337pt}{-2.25pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=2$,even}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\quad\leavevmode\hbox to224.92pt{\vbox to137.47pt{\pgfpicture\makeatletter\hbox{\hskip 112.46071pt\lower-68.73735pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-109.1277pt}{-65.40434pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=130.88284pt]{figs/plott3}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-22.79337pt}{-2.25pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=3$,even}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$
	$\displaystyle\leavevmode\hbox to224.92pt{\vbox to137.47pt{\pgfpicture\makeatletter\hbox{\hskip 112.46071pt\lower-68.73735pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-109.1277pt}{-65.40434pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=130.88284pt]{figs/plott5}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-22.79337pt}{-2.25pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=5$,even}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\quad\leavevmode\hbox to223.48pt{\vbox to137.47pt{\pgfpicture\makeatletter\hbox{\hskip 111.738pt\lower-68.73735pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-108.405pt}{-65.40434pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=130.88284pt]{figs/plott7}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-22.79337pt}{-2.25pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=7$,even}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$
	$\displaystyle\leavevmode\hbox to221.31pt{\vbox to137.47pt{\pgfpicture\makeatletter\hbox{\hskip 110.65396pt\lower-68.73735pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-107.32095pt}{-65.40434pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=130.88284pt]{figs/plotp2}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-65.4725pt}{-2.25pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=2$,even}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\quad\leavevmode\hbox to221.31pt{\vbox to137.47pt{\pgfpicture\makeatletter\hbox{\hskip 110.65396pt\lower-68.73735pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-107.32095pt}{-65.40434pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=130.88284pt]{figs/plotp3}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-65.4725pt}{-2.25pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=3$,even}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},$
	$\displaystyle\leavevmode\hbox to221.31pt{\vbox to137.47pt{\pgfpicture\makeatletter\hbox{\hskip 110.65396pt\lower-68.73735pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-107.32095pt}{-65.40434pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=130.88284pt]{figs/plotp5}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-65.4725pt}{-2.25pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=5$,even}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\quad\leavevmode\hbox to221.31pt{\vbox to137.47pt{\pgfpicture\makeatletter\hbox{\hskip 110.65396pt\lower-68.73735pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-107.32095pt}{-65.40434pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=130.88284pt]{figs/plotp7}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-65.4725pt}{-2.25pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=7$,even}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.$

9B. The length sequence

Let us now also give here the $l_{n}$ for $n\in\{0,\dots,30\}$ (again in order top for $p=2$ to bottom for $p=7$ , excluding $p=\infty$ where the sequence is [OEI23, A001405]) so that it is copy-able:

$\{1,1,3,3,11,11,41,41,155,155,593,593,2289,2289,8891,8891,34683,34683,135697,135697,532041,532041,2089363,2089363,8215553,8215553,32339011,32339011,127417011,127417011,502458289\}$ .

$\{1,1,2,4,7,14,26,50,97,184,364,692,1378,2641,5264,10181,20267,39523,78524,154187,305728,603614,1194758,2368906,4682134,9313411,18387902,36663241,72331456,144466892,28488466\}$ .

$\{1,1,2,3,6,11,21,42,78,161,297,617,1144,2366,4432,9088,17223,34986,67049,135013,261326,522271,1019427,2024828,3979781,7866186,15547861,30614847,60783158,119345091,237790431\}$ .

$\{1,1,2,3,6,10,20,36,71,135,262,517,990,2001,3796,7786,14690,30379,57188,118712,223515,464341,875955,1817598,3439375,7119305,13522875,27902564,53222511,109424657,209629719\}$ .

As before for $b_{n}$ , here is one picture to compare their growth, including the case $p=\infty$ :

\displaystyle\leavevmode\hbox to236.48pt{\vbox to149.04pt{\pgfpicture\makeatletter\hbox{\hskip 118.24231pt\lower-74.51895pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-114.9093pt}{-71.18594pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=142.26378pt]{figs/lengthall}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.

Note the following comparison of the $l_{n}$ plot to the $b_{n}$ plot from above: for $p<p^{\prime}$ , the $l_{n}$ grows a faster for $p$ and slower for $p^{\prime}$ , and vice versa for the $b_{n}$ .

9C. The generating function

Continuing with Section 3A, recall from Equation 1B.2 that $2(t_{2}+1)\approx 0.585$ . We compare $F(w)$ , with the sum cut-off at $k=10$ , with $(w-1)^{-0.585}$ :

The growth rates towards the singularity $w=1$ (left side of the picture) are almost the same and this is crucial in Section 6.

Recall from Section 4B that

\displaystyle\mathbf{F}_{0}(w)=\big{(}\ln(w^{-1})\big{)}^{\log_{p}(\frac{p+1}{2})}\sum_{k=-\infty}^{\infty}\frac{w^{p^{k}}(1-w^{p^{k}(p-1)})}{(1-w^{p^{k}})(1+w^{p^{k+1}})}\Pi(w^{p^{k}})\left(\frac{p+1}{2}\right)^{k}.

We now give formulas to compute the Fourier coefficients $L_{n}$ of $\mathbf{f}_{0}$ , and thus of $\mathbf{F}_{0}$ , (fairly) efficiently. It turns out that $L_{0}$ is moderate but $L_{n}$ are tiny for $n\neq 0$ , which causes the behavior of $\mathbf{F}_{0}(2^{-p^{x}})$ as in the next example, plotted on $[0.1,0.9]$ :

\displaystyle\leavevmode\hbox to227.09pt{\vbox to143.26pt{\pgfpicture\makeatletter\hbox{\hskip 113.54475pt\lower-71.62816pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-110.21175pt}{-68.29515pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=136.5733pt]{figs/fzero}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{5.38959pt}{19.08957pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=2$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.

This looks like the sine curve because $\sum_{n>1}|L_{n}|$ is much smaller than $|L_{1}|$ as we will see momentarily. To wrap-up, for $p=2$ one can show that

$\displaystyle L_{n}=\frac{1}{(\ln 2)^{2-\log_{2}(3/2)}}\int_{0}^{1}\frac{\ln(w^{-1})^{\log_{2}(\frac{3}{2})+in\log_{2}(e^{2\pi})}}{w+w^{-1}}\prod_{j=1}^{\infty}\left(1-\frac{(w^{2^{-j-1}}-w^{-2^{-j-1}})^{2}}{3(w^{2^{-j}}+w^{-2^{-j}})}\right)\frac{dw}{w\log_{2}w^{-1}}.$

Let $\zeta(x,s)=\sum_{k=0}^{\infty}(k+x)^{-s}$ denote the Hurwitz zeta function. Consider the following reparametrization:

\displaystyle\xi(\beta,u)=4^{-\beta}\zeta(\beta,\tfrac{u}{4}).

Recalling that $\Gamma(z)=\int_{0}^{\infty}x^{z-1}e^{-x}dx=\int_{0}^{1}(\ln\frac{1}{x})^{z-1}dx$ , we get

\displaystyle L_{n}=\frac{2\Gamma\big{(}\log_{2}(\tfrac{3}{2})+1+\tfrac{2\pi in}{\ln 2}\big{)}}{(\ln 2)^{1+\frac{2\pi in}{\ln 2}}}\lim_{N\to\infty}\frac{1}{3^{N}}\sum_{-1\leq k_{1},\dots,k_{N}\leq 1}\xi\big{(}\log_{2}(\tfrac{3}{2})+1+\tfrac{2\pi in}{\ln 2},2+{\textstyle\sum_{j=1}^{N}}k_{j}2^{-j}\big{)}.

The Hurwitz zeta function is quite easy to compute, so this gives a fairly efficient way to compute the Fourier coefficients $L_{n}$ .

A bit of work shows that, for general $p$ , we have:

$L_{n}=\frac{2\Gamma(\log_{p}(\frac{p+1}{2})+1+\frac{2\pi in}{\ln p})}{(\ln p)}\lim_{N\to\infty}\frac{1}{(p+1)^{N}}\sum_{-1\leq k_{1},\dots,k_{N}\leq 1}\xi(\log_{p}(\tfrac{p+1}{2})+1+\tfrac{2\pi in}{\ln p},2+{\textstyle\sum_{j=1}^{N}}k_{j}p^{-j}).$

Moreover, using the classical formulas for the Hurwitz zeta function one can show that $|L_{n}|$ is nonzero. Furthermore, let

\displaystyle S(p,n,N):=\bigg{|}\sum_{-1\leq k_{1},\dots,k_{N}\leq 1}\xi(\log_{p}(\tfrac{p+1}{2})+1+\tfrac{2\pi in}{\ln p},2+{\textstyle\sum_{j=1}^{N}}k_{j}p^{-j})\bigg{|}.

One can also show that $\max\{S(p,n,N)\}$ , for fixed $N$ , is obtained at $p=2$ and $n=0$ . Indeed, the function $S(p,n,N)$ behaves as follows when varying $n$ and $p$ , respectively, while keeping $N$ fixed:

\displaystyle\leavevmode\hbox to217.69pt{\vbox to143.26pt{\pgfpicture\makeatletter\hbox{\hskip 108.8472pt\lower-71.62816pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-105.51419pt}{-68.29515pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=136.5733pt]{figs/fourier1}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{15.0456pt}{-2.44444pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=2,N=5$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\quad\leavevmode\hbox to217.69pt{\vbox to143.26pt{\pgfpicture\makeatletter\hbox{\hskip 108.8472pt\lower-71.62816pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-105.51419pt}{-68.29515pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=136.5733pt]{figs/fourier2}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{14.56004pt}{-2.44444pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$n=0,N=5$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.

Here we have divided by $3^{N}$ to highlight the bound $0.6$ which is quite crude:

\displaystyle\leavevmode\hbox to228.53pt{\vbox to143.26pt{\pgfpicture\makeatletter\hbox{\hskip 114.26746pt\lower-71.62816pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-110.93445pt}{-68.29515pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=136.5733pt]{figs/fourier3}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{16.6069pt}{-2.25pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=2,n=0$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.

9D. Numerical data for odds and ends

Recall that Section 1B shows that, for some $C_{1},C_{2}\in\mathbb{R}_{>0}$ , we have

\displaystyle C_{1}\cdot n^{t_{p}}\cdot 2^{n}\leq b_{n}\leq C_{2}\cdot n^{t_{p}}\cdot 2^{n},\qquad n\geq 1.

One can probably take $C_{1}=1/4$ and $C_{2}=1$ , as illustrated here for $p=2$ :

\displaystyle\leavevmode\hbox to219.86pt{\vbox to149.04pt{\pgfpicture\makeatletter\hbox{\hskip 109.93126pt\lower-74.51895pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-106.59825pt}{-71.18594pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=142.26378pt]{figs/lowerupperp2a}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.36137pt}{-2.25pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=2$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\leavevmode\hbox to224.2pt{\vbox to149.04pt{\pgfpicture\makeatletter\hbox{\hskip 112.09935pt\lower-74.51895pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-108.76634pt}{-71.18594pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=142.26378pt]{figs/lowerupperp2b}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.36137pt}{-2.25pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=2$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.

Moreover, Section 5A shows that we have

\displaystyle b_{n+2}\leq 4b_{n},\;n\geq 0.

One could conjecture that $\lim_{n\to\infty}b_{n+2}/b_{n}=4$ , as illustrated here for $p=2$ where we know this is true by Section 7A:

\displaystyle\leavevmode\hbox to225.64pt{\vbox to136.03pt{\pgfpicture\makeatletter\hbox{\hskip 112.82205pt\lower-68.01465pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-109.48904pt}{-64.68164pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=129.46011pt]{figs/monoton1}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.36137pt}{-2.25pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=2$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\leavevmode\hbox to225.64pt{\vbox to136.03pt{\pgfpicture\makeatletter\hbox{\hskip 112.82205pt\lower-68.01465pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-109.48904pt}{-64.68164pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=129.46011pt]{figs/monoton2.png}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-54.36137pt}{-2.25pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=2$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.

Finally, $W=\mathrm{Sym}^{2}V$ with $\dim_{\mathbf{k}}W=3$ is an indecomposable tilting $SL_{2}(\mathbf{k})$ -representation unless $p=2$ . For $p=3$ we get for $b_{n}=b_{n}(W)$ the numbers:

	$\displaystyle(b_{n})_{n\in\mathbb{Z}_{\geq 0}}=($	$\displaystyle 1,1,2,5,13,35,95,260,715,1976,5486,15301,$
		$\displaystyle 42686,120628,340874,967136,2754455,7872973,\dots).$

One could aim for a statement of the form

\displaystyle D_{1}\cdot n^{t_{p}}\cdot 3^{n}\leq b_{n}\leq D_{2}\cdot n^{t_{p}}\cdot 3^{n},\qquad n\geq 1,

where $D_{1},D_{2}\in\mathbb{R}_{>0}$ . Indeed, we get the following logplots:

\displaystyle\leavevmode\hbox to236.48pt{\vbox to143.26pt{\pgfpicture\makeatletter\hbox{\hskip 118.24231pt\lower-71.62816pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-114.9093pt}{-68.29515pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=136.5733pt]{figs/sym1}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{30.9969pt}{-2.25pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=3$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\leavevmode\hbox to228.53pt{\vbox to143.26pt{\pgfpicture\makeatletter\hbox{\hskip 114.26746pt\lower-71.62816pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ } {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{-110.93445pt}{-68.29515pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{\includegraphics[height=136.5733pt]{figs/sym2}}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} {{}}\hbox{\hbox{{\pgfsys@beginscope\pgfsys@invoke{ }{{}{}{{ {}{}}}{ {}{}} {{}{{}}}{{}{}}{}{{}{}} { }{{{{}}\pgfsys@beginscope\pgfsys@invoke{ }\pgfsys@transformcm{1.0}{0.0}{0.0}{1.0}{30.9969pt}{-2.25pt}\pgfsys@invoke{ }\hbox{{\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\hbox{{$p=3$}} }}\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope}}} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{ {}{}{}{}{}}{{{}}{{}}}}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}.

As one can see, we expect some nontrivial scalar to make the curves fit better. Hence, these plots might indicate an analog of Section 1B for $W$ instead of $V$ .

Appendix A Monoidal subcategories of finite tensor categories

We use the terminology of tensor categories from [EGNO15]. Recall that an object $X$ of a tensor category $\mathbf{C}$ is a generator if any object of $\mathbf{C}$ is isomorphic to a subquotient of a direct sum of tensor products of $X$ and its (iterated) duals.

For Section 2A, we need to show that in case $\mathbf{C}$ is finite (meaning that $\mathbf{C}$ has enough projective objects and finitely many simple objects), all projective objects appear as direct summands of the tensor powers of $X$ . This follows from Appendix A.(d) below and the fact that projective objects in a tensor category are also injective; see [EGNO15, Proposition 6.1.3]. Note that the theorem is a generalization of the known case of fusion categories in [DGNO10, Corollary F.7]. It can also be interpreted as a generalization of the Burnside–Brauer–Steinberg theorem for Hopf algebras; see, e.g. [PQ95], or a generalization of the main result of [BK72].

Theorem \theTheorem.

The following properties hold for any tensor category $\mathbf{C}$ with finitely many isomorphism classes of simple objects:

(a)

Any full additive subcategory that is closed under taking subquotients and internal tensor products is a tensor subcategory (i.e. rigid).
(b)

For any $X\in\mathbf{C}$ , the left and right dual objects $X^{\ast}$ and ${}^{\ast}X$ appear as subquotients of direct sums of $X^{\otimes d}$ for $d\in\mathbb{Z}_{\geq 0}$ .
(c)

For any simple object $V\in\mathbf{C}$ , the left and right dual objects $V^{\ast}$ and ${}^{\ast}V$ appear as subquotients of $V^{\otimes d}$ for some $d\in\mathbb{Z}_{\geq 0}$ .
(d)

If $X$ is a generator, then every object in $\mathbf{C}$ is a subquotient of direct sums of tensor powers of $X$ .

Proof.

Firstly, we can observe that part (a) implies part (b) and that (b) implies (d).

Now we prove that (c) actually implies (a). For this, let $\mathbf{D}\subset\mathbf{C}$ be a subcategory as in (a), and we assume that property (c) is true for $\mathbf{C}$ . To show that $\mathbf{D}$ is rigid, we show that every object $Z\in\mathbf{D}$ is rigid. We prove this by induction on the length $\ell(Z)$ of $Z\in\mathbf{D}$ . The base $\ell(Z)=1$ follows from our assumption (c).

To justify the induction step, let $\ell(Z)\geq 2$ and include $Z$ into a short exact sequence

\displaystyle 0\to Y\to Z\to A\to 0,

where $Y,A$ are nonzero. Note that $A$ and $Y$ are rigid in $\mathbf{D}$ by induction, and $Z$ is defined by an element $h\in\mathrm{Ext}(A,Y):=\mathrm{Ext}_{\mathbf{D}}(A,Y)$ . Now, in general, given an element $h\in\mathrm{Ext}(A,Y)$ for objects with duals $A$ and $Y$ , we have an element $1\otimes h\otimes 1\in\mathrm{Ext}(Y^{\ast}\otimes A\otimes A^{\ast},Y^{\ast}\otimes Y\otimes A^{\ast})$ which defines an element $h^{\ast}\in\mathrm{Ext}(Y^{\ast},A^{\ast})$ obtained by composing $1\otimes h\otimes 1$ with evaluations and coevaluations. The $Z^{\ast}$ , the extension of $A^{\ast}$ and $Y^{\ast}$ defined by $h^{\ast}$ is the left dual of $Z$ , and thus belongs to $\mathbf{D}$ . The right dual ${}^{\ast}Z$ can be found in a similar way. Thus, $Z$ is rigid.

By the above, we have (c) $\Rightarrow$ (a) $\Rightarrow$ (b) $\Rightarrow$ (d), and it now suffices to prove property (c). This will be done below after some preparation.

The following lemma can be seen as being part of the Frobenius–Perron theorem.

Lemma \theLemma.

Let $M$ be a nonzero square matrix with nonnegative entries and $v,w$ be row and column vectors with strictly positive entries such that $vM=\mu_{1}v$ , $Mw=\mu_{2}w$ for some $\mu_{1},\mu_{2}\in\mathbb{R}$ . Then $\mu_{1}=\mu_{2}>0$ and $M$ is completely reducible (conjugate by a permutation matrix to a direct sum of irreducible matrices).

Proof.

We have $\mu_{1}vw=\mu_{2}vw=vMw>0$ , so $\mu_{1}=\mu_{2}=\mu>0$ . It remains to show that if $M$ is reducible, then it is decomposable. If $M$ is reducible, then after conjugating $M$ by a permutation matrix, we have $M=\begin{pmatrix}M_{11}&M_{12}\\ 0&M_{22}\end{pmatrix}$ . So, if $v=(v_{1},v_{2})$ , $w=\binom{w_{1}}{w_{2}}$ then we have

\displaystyle M_{22}w_{2}=\mu w_{2},\ v_{1}M_{12}+v_{2}M_{22}=\mu v_{2}.

Multiplying the second equation by $w_{2}$ , the first equation by $v_{2}$ , and subtracting, we get $v_{1}M_{12}w_{2}=0$ , which means $M_{12}=0$ , so $M$ is decomposable, as claimed. ∎

Now, let $A$ be a transitive unital $\mathbb{Z}_{+}$ -ring of finite rank with basis $B=(b_{j})$ . For $X:=\sum x_{j}b_{j}$ with $x_{j}\in\mathbb{R}$ , let $M_{X}$ be the matrix of left multiplication by $X$ on $A\otimes_{\mathbb{Z}}\mathbb{R}$ in the basis $B$ .

Lemma \theLemma.

(i)

If $x_{j}\geq 0$ , then $M_{X}$ is completely reducible.
(ii)

Suppose that $b,b^{\ast}\in B$ and that the $B$ -decomposition of $bb^{\ast}$ contains $b_{0}=1$ . Then there exists $n\in\mathbb{Z}_{\geq 0}$ such that the $B$ -decomposition of $b^{n}$ contains $b^{\ast}$ .

Proof.

(i). Let $d\colon A\to\mathbb{R}$ be the Frobenius–Perron dimension. Since $d$ is a character, $d(X)d(b_{j})=d(Xb_{j})=\sum_{k}(M_{X})_{jk}d(b_{k})$ , so the column vector $(d(b_{k}))$ is a right eigenvector of $M_{X}$ . Also, by [EGNO15, Proposition 3.3.6(2)], there exists a left (row) eigenvector $(c_{k})$ of $M_{X}$ with positive entries. Thus, by Appendix A, $M_{X}$ is completely reducible.

(ii). Consider the directed graph $\Gamma$ with vertex set $B$ and edge $b_{i}\to b_{j}$ present if and only if the $B$ -decomposition of $bb_{i}$ contains $b_{j}$ . Let $\Gamma_{1}$ be the connected component of $\Gamma$ that contains $b_{0}=1$ . Since the $B$ -decomposition of $bb^{\ast}$ contains $1$ , $b^{\ast}\in\Gamma_{1}$ . Thus, by (i), there is an oriented path from $1$ to $b^{\ast}$ , i.e., there exists $n\in\mathbb{Z}_{\geq 0}$ such that the $B$ -decomposition of $b^{n}$ contains $b^{\ast}$ , as desired. ∎

To complete the proof of Appendix A, it now suffices to observe that Appendix A.(c) follows from applying Appendix A(ii) to the Grothendieck ring $A=K_{0}(\mathbf{C})$ , with $B$ the basis of simple objects, $b=[V]$ and $b^{\ast}=[V^{\ast}]$ or $b^{\ast}=[{}^{\ast}V]$ . ∎

Remark \theRemark.

The assumption that $\mathbf{C}$ has finitely many simple objects in Appendix A cannot be dropped (even for categories with enough projective objects). For instance, the statement is not true if $\mathbf{C}$ is the representation category of the multiplicative group and $X$ is a faithful one dimensional representation.

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Fractal behavior of tensor powers of the two dimensional space in prime characteristic

Abstract.

Key words and phrases:

2020 Mathematics Subject Classification:

1. Introduction

1A. The exponential factor

1B. The main theorem – the subexponential factor

Example \theExample.

Example \theExample.

Remark \theRemark.

Notation \theNotation.

Main Theorem \theMain.Theorem.

Remark \theRemark.

Remark \theRemark.

Example \theExample.

Remark \theRemark.

Remark \theRemark.

1C. Proof outline of Section 1B

1D. Odds and ends

2. Fractal behavior of growth problems

2A. No fractal behavior

Proposition \theProposition.

Proposition \theProposition.

Example \theExample.

Remark \theRemark.

2B. Fractal behavior

Remark \theRemark.

2BI. Modular representations and the Cantor set

Remark \theRemark.

2BII. Generalization to general primes and highest weights

2BIII. Generalization to higher rank

2BIV. Modular representations and Sierpinski gaskets

Remark \theRemark.

2BV. Lengths of tensor powers

Lemma \theLemma.

Proof.

2BVI. Conclusion and goal

3. Generating function

3A. The function FF

Theorem \theTheorem.

Remark \theRemark.

Example \theExample.

Example \theExample.

3B. Recursion relations

Example \theExample.

Lemma \theLemma.

Proof.

3C. Functional equation

Proposition \theProposition.

Example \theExample.

Proof of Section 3C.

3D. Conclusion

Proof of Section 3A.

4. Asymptotics of the generating function

4A. Asymptotics

Theorem \theTheorem.

Proof.

4B. The oscillating factor

Lemma \theLemma.

Proof.

Proposition \theProposition.

Remark \theRemark.

Proof of Section 4B.

Proposition \theProposition.

Proof.

4C. Example for p=2p=2

Lemma \theLemma.

Proof.

Proposition \theProposition.

Proof.

5. Monotonicity

5A. Neighboring values of bnb_{n}

Theorem \theTheorem.

Proof.

Lemma \theLemma.

Proof.

Lemma \theLemma.

Proof.

Lemma \theLemma.

Example \theExample.

3A. The function $F$

4C. Example for $p=2$

5A. Neighboring values of $b_{n}$

5B. Example for $p=2$

7A. Neighboring values of $b_{n}$ in characteristic two

8C. Analysis of the Fourier coefficients of the generating function for $b_{n}/2^{n}$