Extended-cycle integrals of modular functions for badly approximable numbers

Yuya Murakami Mathematical Inst. Tohoku Univ., 6-3, Aoba, Aramaki, Aoba-Ku, Sendai 980-8578, JAPAN [email protected]

Abstract.

Cycle integrals of modular functions are expected to play a role in real quadratic analogue of singular moduli. In this paper, we extend the definition of cycle integrals of modular functions from real quadratic numbers to badly approximable numbers. We also give explicit representations of values of extended-cycle integrals for some cases.

The author is supported by JSPS KAKENHI Grant Number JP 20J20308.

1. Introduction

The elliptic modular $j$ -function $j(z)$ is an $\operatorname{SL}_{2}(\mathbb{Z})$ -invariant holomorphic function on the upper half-plane $\mathbb{H}:=\{z=x+y\sqrt{-1}\in\mathbb{C}\mid x,y\in\mathbb{R},y>0\}$ . It plays an essential role in the complex multiplication theory. Indeed, its special values at imaginary quadratic points generate the Hilbert class fields of imaginary quadratic fields. On the other hand, for real quadratic fields, any definitive construction of the Hilbert class fields or ray class fields is still unknown except for a few cases, that is [Shi72], [Shi78]. With this motivation, Kaneko [Kan09] introduced the “value” of $j(z)$ , written $\operatorname{val}(w)$ derived from Dedekind’s notation, at any real quadratic number $w$ which is defined by

(1.1)

\operatorname{val}(w):=\frac{1}{2\log\varepsilon_{w}}\int_{z_{0}}^{\gamma_{w}z_{0}}j(z)\bigg{(}\dfrac{1}{z-w^{\prime}}-\dfrac{1}{z-w}\bigg{)}dz,

where $z_{0}\in\mathbb{H}$ is any point, $w^{\prime}$ is the non-trivial Galois conjugate of $w$ , $\operatorname{SL}_{2}(\mathbb{Z})_{w}$ is the stabilizer of $w$ in $\operatorname{SL}_{2}(\mathbb{Z})$ , and $\gamma_{w}=\begin{pmatrix}*&*\\ c&d\end{pmatrix}$ is the unique element of $\operatorname{SL}_{2}(\mathbb{Z})_{w}$ such that $\operatorname{SL}_{2}(\mathbb{Z})_{w}=\{\pm\gamma_{w}^{n}\mid n\in\mathbb{Z}\}$ and $\varepsilon_{w}:=cw+d>1$ . The integral is independent of $z_{0}$ since the integrand is holomorphic and invariant under $\gamma_{w}$ . We can prove that $\varepsilon_{w}$ is the minimal unit greater than $1$ with the norm $1$ in the order $\mathcal{O}_{w}$ in the real quadratic field $\mathbb{Q}(w)$ whose discriminant coincides with the discriminant of $w$ .

This value can be written as $\operatorname{val}(w)=\widetilde{\operatorname{val}}(w)/\widetilde{1}(w)$ , where

(1.2)

\widetilde{\operatorname{val}}(w):=\int_{\operatorname{SL}_{2}(\mathbb{Z})_{w}\backslash S_{w^{\prime},w}}jds,\quad\widetilde{1}(w):=\int_{\operatorname{SL}_{2}(\mathbb{Z})_{w}\backslash S_{w^{\prime},w}}ds=2\log\varepsilon_{w}

are cycle integrals, $ds:=y^{-1}\sqrt{dx^{2}+dy^{2}}$ , and $S_{w^{\prime},w}$ is the geodesic in $\mathbb{H}$ from $w^{\prime}$ to $w$ .

In the last decade, it turned out that the values of $\widetilde{\operatorname{val}}$ have interesting properties. They are related to a mock modular form of weight $1/2$ for $\Gamma_{0}(4)$ studied by Duke–Imamoglu–Tóth [DIT11], which is a real quadratic analogue of Zagier’s work on traces of singular moduli in [Zag02]. They are also related to a locally harmonic Maass form of weight $2$ studied by Matsusaka [Mat20, Theorem 1.1]. Bengoechea–Imamoglu [BI19, Theorem 1.1] showed some kind of continuity for the values of $\operatorname{val}$ at Markov quadratics which is conjectured by Kaneko [Kan09]. The author [Mur20, Theorem 1.1] generalized it to the values of $\operatorname{val}$ at real quadratic numbers whose periods of continued fraction expansions have long cyclic parts. It revealed that the continuity of $\operatorname{val}$ differs from Euclidean topology. However, the definition of the continuity of $\operatorname{val}$ is still missing. We would expect a topological space $H$ containing all real quadratic numbers and a continuous function $?\colon H\to\mathbb{C}$ such that the diagram

(1.3)

commutes.

In this paper, we attempt to choose $H$ as a set of badly approximable numbers and give some partial continuity of $\operatorname{val}$ on $H$ . Here, badly approximable numbers are defined as irrational numbers $x$ whose regular continued fraction expansions

(1.4)

x=[k_{1},k_{2},k_{3},\dots]:=k_{1}+\cfrac{1}{k_{2}+{\cfrac{1}{k_{3}+\cfrac{1}{\ddots}}}},\quad k_{1}\in\mathbb{Z},\quad k_{2},k_{3},\dots\in\mathbb{Z}_{>0}

have bounded coefficients, that is, the set $\{k_{1},k_{2},k_{3},\dots\}$ is bounded. For example, real quadratic irrational numbers are badly approximable since they have periodic continued fraction expansions.

We define $\operatorname{val}(x)$ and related values for a badly approximable number $x=[k_{1},k_{2},k_{3},\dots]$ as

(1.5)	$\displaystyle\operatorname{val}(x)$	$\displaystyle:=\lim_{\begin{subarray}{c}z\in S_{z_{0},x}\\ z\to x\end{subarray}}\frac{1}{d_{\mathrm{hyp}}(z_{0},z)}\int_{S_{z_{0},z}}jds,$
(1.6)	$\displaystyle\widehat{\operatorname{val}}(x)$	$\displaystyle:=\lim_{n\to\infty}\frac{1}{n}\int_{z_{0}}^{\gamma_{n}z_{0}}j(z)\frac{-1}{z-x}dz,$
(1.7)	$\displaystyle\widehat{1}(x)$	$\displaystyle:=\lim_{n\to\infty}\frac{1}{n}\int_{z_{0}}^{\gamma_{n}z_{0}}\frac{-1}{z-x}dz,$
(1.8)	$\displaystyle\widehat{\varepsilon}_{x}$	$\displaystyle:=\lim_{n\to\infty}c_{n}^{1/n},$

where $z_{0}\in\mathbb{H}$ , $S_{z_{0},z}$ is the geodesic in $\mathbb{H}$ from $z_{0}$ to $z\in\mathbb{H}\cup\mathbb{R}$ ,

(1.9)

d_{\mathrm{hyp}}(z_{0},z):=\int_{S_{z_{0},z}}ds,\quad\gamma_{n}:=\begin{pmatrix}k_{1}&1\\ 1&0\end{pmatrix}\begin{pmatrix}k_{2}&1\\ 1&0\end{pmatrix}\dots\begin{pmatrix}k_{2n}&1\\ 1&0\end{pmatrix}\in\operatorname{SL}_{2}(\mathbb{Z}),

and $c_{n}\in\mathbb{Z}_{>0}$ be the denominator of the rational number $[k_{1},k_{2},\cdots k_{2n}]$ . In Section 3 later, we will introduce more general representations of these values.

We will prove that these limit values are bounded, but we do not know if the limit values are singleton. Hence, the right-hand sides of the above equations do converge for any badly approximable number. We can also prove that if $x$ is a real quadratic number, then the limits $\operatorname{val}(x),\widehat{\operatorname{val}}(x),\widehat{1}(x)$ , and $\widehat{\varepsilon}_{x}$ converge, $\operatorname{val}(x)$ coincides with the values defined in 1.1, and we can write

(1.10)

\widehat{\operatorname{val}}(x)=\frac{\widetilde{\operatorname{val}}(x)}{r_{x}},\quad\widehat{1}(x)=\frac{\widetilde{1}(x)}{r_{x}},\quad\widehat{\varepsilon}_{x}=\varepsilon_{x}^{1/r_{x}},

where $r_{x}$ are half of the length of the minimum even period of the continued fraction expansion of $x$ . Thus, $\widehat{\varepsilon}_{x}$ for a badly approximable number $x$ is an analog of fundamental units of real quadratic orders.

Our purpose in this paper is to give uncountable infinitely many badly approximable numbers $x$ such that the limits $\operatorname{val}(x),\widehat{\operatorname{val}}(x),\widehat{1}(x),$ and $\varepsilon_{x}$ converge. We also explicitly represent values of extended $\operatorname{val}$ function at badly approximable numbers which have infinitely long cyclic parts in their continued fractions.

To explain the main results, we introduce some notation for even words. For an even number $2r\in 2\mathbb{Z}_{\geq 0}$ and positive integers $k_{1},\dots,k_{2r}$ , we call a tuple $W=(k_{1},\dots,k_{2r})$ an even word. In the case when $r=0$ , we call it the empty word and denote it by $\emptyset$ . For even words $W_{1}=(k_{1}^{(1)},\dots,k_{2r_{1}}^{(1)}),\dots,W_{n}=(k_{1}^{(n)},\dots,k_{2r_{n}}^{(n)})$ , define their product

(1.11)

W_{1}\cdots W_{n}:=(k_{1}^{(1)},\dots,k_{2r_{1}}^{(1)},\dots,k_{1}^{(n)},\dots,k_{2r_{n}}^{(n)}).

For an infinite sequence of even words $W_{1}=(k_{1}^{(1)},\dots,k_{2r_{1}}^{(1)}),\dots,W_{n}=(k_{1}^{(n)},\dots,k_{2r_{n}}^{(n)}),\dots$ , define

(1.12)

[W_{1}\cdots W_{n}\cdots]:=[k_{1}^{(1)},\dots,k_{2r_{1}}^{(1)},\dots,k_{1}^{(n)},\dots,k_{2r_{n}}^{(n)},\dots].

For a non-empty even word $W$ , define

(1.13)

\displaystyle N(W)

\displaystyle:=\max\{n\in\mathbb{Z}_{>0}\mid\text{ there exists an even word }W_{1}\text{ such that }W=W_{1}^{n}\}.

Our first main result is as follows.

Theorem 1.1.

Let $V_{0},\dots,V_{k}$ be even words, $W_{1},\dots,W_{k}$ be non-empty even words, and $\{a_{1,n}\}_{n=1}^{\infty}$ , $\dots$ , $\{a_{k,n}\}_{n=1}^{\infty}$ be sequences in $\mathbb{Z}_{\geq 0}$ such that $a_{i,n}\to\infty$ and $2^{-n}a_{i,n}\to 0$ when $n$ goes to infinity. Let $w_{i}:=[\overline{W_{i}}]$ , $k^{\prime}:=\{0\leq i\leq k\mid V_{i}\neq\emptyset\}$ ,

(1.14)

A_{n}:=k^{\prime}n+\sum_{1\leq i\leq k}\sum_{1\leq j\leq n}a_{i,j},\quad a_{i}:=\lim_{n\to\infty}\frac{1}{A_{n}}\sum_{1\leq j\leq n}a_{i,j},

and

(1.15)

U_{n}:=V_{1}W_{1}^{a_{1,n}}V_{2}W_{2}^{a_{2,n}}\cdots V_{k}W_{k}^{a_{k,n}},\quad x:=[U_{1}U_{2}\cdots U_{n}\cdots].

Then the limits $\operatorname{val}(x),\widehat{\operatorname{val}}(x),\widehat{1}(x),$ and $\widehat{\varepsilon}_{x}$ converge and it holds $\widehat{1}(x)=2\log\widehat{\varepsilon}_{x}>0$ and $\operatorname{val}(x)=\widehat{\operatorname{val}}(x)/\widehat{1}(x)$ . Moreover, it also holds

(1.16)

\operatorname{val}(x)=\frac{a_{1}N(W_{1})\widehat{\operatorname{val}}(w_{1})+\dots+a_{k}N(W_{k})\widehat{\operatorname{val}}(w_{k})}{a_{1}N(W_{1})\widehat{1}(w_{1})+\dots+a_{k}N(W_{k})\widehat{1}(w_{k})}.

Here we remark that the author proved that the right-hand side in 1.16 equals to the limit of $\operatorname{val}([\overline{U_{n}}])$ in [Mur20, Theorem 1.1].

Next, we consider the value of $\operatorname{val}$ function at the Thue–Morse word. It is a typical word which is repetition-free. We fix two even words $V$ and $W$ . Let $\{V,W\}^{\omega}$ be the set of even words generated by $V,W$ which are of finite length. Let $\{V,W\}^{\omega}$ be the set of even words generated by $V,W$ which are of infinite length. We define the monoid homomorphism $h\colon\{V,W\}^{*}\to\{V,W\}^{*}$ by $h(V):=VW,h(W):=WV$ . In this setting, the word

(1.17)

V_{h}:=\lim_{n\to\infty}h^{n}(V)=VWWVWVVW\cdots\in\{V,W\}^{\omega}

is called the Thue–Morse word. It is known that $V_{h}$ is cubefree (for example, see [CK97, Section 3]), and thus it is not a word which satisfies the conditions in Theorem 1.1.

The value of $\operatorname{val}$ function at the Thue–Morse word is calculated as follows.

Theorem 1.2.

Let $x:=[V_{h}]$ and $w_{n}:=[\overline{h^{n}(V)}]$ . Then $\widehat{\operatorname{val}}(x)$ converges if and only if $\lim_{n\to\infty}\widehat{\operatorname{val}}(w_{2n})/2^{2n}$ converges. Moreover, we have

(1.18)

\widehat{\operatorname{val}}(x)=\lim_{n\to\infty}\frac{1}{2^{2n}}\widehat{\operatorname{val}}(w_{2n}),\quad\widehat{1}(x)=\lim_{n\to\infty}\frac{1}{2^{2n}}\widehat{1}(w_{2n}),\quad\operatorname{val}(x)=\lim_{n\to\infty}\frac{1}{2^{2n}}\operatorname{val}(w_{2n}).

The definitions of $\widehat{\operatorname{val}}(x)$ and $\widehat{1}(x)$ are like the Birkoff average in ergodic theory. Khinchin-Lévy’s theorem in ergodic theory asserts that for almost all $x\in[0,1]$ with respect to the measure

(1.19)

d\mu:=\frac{1}{\log 2}\frac{dx}{1+x}

on $[0,1]$ , the limit defining $\widehat{\varepsilon}_{x}$ converges and we have

(1.20)

\log\widehat{\varepsilon}_{x}=-2\int_{0}^{1}\log(x)d\mu=\frac{\pi^{2}}{6\log 2}.

Although ergodic theory makes it possible to study values like the Birkoff average at almost all points, it does not tell us about a value at each point, for example, a badly approximable number. We can state that this paper studies a range which ergodic theory does not deal with.

This paper will be organized as follows. In Section 2, we give renewal definitions of $\operatorname{val}(w),\widehat{\operatorname{val}}(w)$ , and $\widehat{1}(w)$ for real quadratic numbers $w$ , which are suitable to generalize for badly approximable numbers. In Section 3, we state some fundamental properties of the limits $\operatorname{val}(x),\widehat{\operatorname{val}}(x),\widehat{1}(x),$ and $\widehat{\varepsilon}_{x}$ for each badly approximable number $x$ . In Section 4, we describe $\operatorname{val}(x)$ in terms of the limit value along a geodesic and study its properties. In Section 5, we also study basic properties of $\widehat{\operatorname{val}}(x)$ and $\widehat{1}(x)$ which are defined using the continued fraction expansion of $x$ . In Section 6, we study $\varepsilon_{x}$ and a relation between it and $\widehat{1}(x)$ . Finally we prove Theorem 1.1 in Section 7 and Theorem 1.2 in Section 8.

Acknowledgement

I would like to show my greatest appreciation to Professor Takuya Yamauchi for giving many pieces of advice. I am deeply grateful to Dr. Toshiki Matsusaka for giving many comments. I would like to express my gratitude to Professor Shun’ichi Yokoyama and Dr. Toshihiro Suzuki for giving me constructive comments on computing $\operatorname{val}(x)$ numerically. It is a pleasure to extend my thanks to Professor Tatsuya Tate for teaching me ergodic theory. I also thank Dr. Daisuke Kazukawa, Dr. Hiroki Nakajima, and Dr. Shin’ichiro Kobayashi for teaching me geodesics, hyperbolic geometry, and metric spaces. I appreciate the technical assistance of Dr. Naruaki Kato for introducing me to how to write works by using GitHub.

2. A renewal definition of $\operatorname{val}$ for real quadratic numbers

In this section, we give renewal definitions of $\operatorname{val}$ for real quadratic numbers $w$ , which are suitable to generalize for badly approximable numbers. To begin with, we list some basic notations and facts.

2.1. Basic notation and properties for geodesics

Let $\mathbb{H}:=\{z=x+y\sqrt{-1}\in\mathbb{C}\mid x,y\in\mathbb{R},y>0\}$ be the upper half plane and $ds:=y^{-1}\sqrt{dx^{2}+dy^{2}}$ be its line element. Here we remark that $ds$ is invariant under $\operatorname{SL}_{2}(\mathbb{R})$ . We denote by $\infty:=\lim_{t\to+\infty}t\sqrt{-1}$ the point at infinity. Let $\mathbb{P}^{1}(\mathbb{R}):=\mathbb{R}\cup\{\infty\}$ . For two points $x^{\prime},x\in\mathbb{H}\cup\mathbb{P}^{1}(\mathbb{R})$ , we denote by $S_{x^{\prime},x}$ the geodesic in $\mathbb{H}$ from $x^{\prime}$ to $x$ . For two points $z,z^{\prime}\in\mathbb{H}$ , define hyperbolic distance between them by

(2.1)

d_{\mathrm{hyp}}(z,z^{\prime}):=\int_{S_{z,z^{\prime}}}ds.

The following lemma is fundamental.

Lemma 2.1.

For two points $z,z^{\prime}\in\mathbb{H}$ , the following statements hold.

(i)

For any $\sigma\in\operatorname{SL}_{2}(\mathbb{R})$ , it holds $d_{\mathrm{hyp}}(\sigma(z),\sigma(z^{\prime}))=d_{\mathrm{hyp}}(z,z^{\prime})$ .

(ii)

([Bea83, Theorem 7.2.1 (ii)])

(2.2)

\cosh d_{\mathrm{hyp}}(z,z^{\prime})=1+\frac{\left\lvert z-z^{\prime}\right\rvert^{2}}{2\operatorname{Im}(z)\operatorname{Im}(z^{\prime})}.

For $x^{\prime},x\in\mathbb{P}^{1}(\mathbb{R})$ , define a $1$ -form

(2.3)

\eta_{x^{\prime},x}(z):=\bigg{(}\dfrac{1}{z-x^{\prime}}-\dfrac{1}{z-x}\bigg{)}dz.

Here, let $1/(z-\infty):=0$ if $x=\infty$ or $x^{\prime}=\infty$ .

By a direct calculation, we have the following lemma.

Lemma 2.2 ([Mur20, Lemma 2.5]).

For any $x,x^{\prime}\in\mathbb{P}^{1}(\mathbb{R})$ and $\gamma\in\operatorname{SL}_{2}(\mathbb{R})$ , we have

(2.4)

\gamma^{*}\eta_{x^{\prime},x}=\eta_{\gamma^{-1}x^{\prime},\gamma^{-1}x}.

We need the following lemma to express $\operatorname{val}(w)$ as a quotient of two cycle integrals.

Lemma 2.3.

For two points $x,x^{\prime}\in\mathbb{P}^{1}(\mathbb{R})$ with $x\neq x^{\prime}$ , we have $ds=\eta_{x^{\prime},x}$ on $S_{x^{\prime},x}$ .

Proof.

Let

(2.5)

\sigma:=\begin{cases}\begin{pmatrix}x&x^{\prime}/(x-x^{\prime})\\ 1&1/(x-x^{\prime})\end{pmatrix}\quad&\text{if }x\neq\infty,\\ \begin{pmatrix}1&-x\\ 0&1\end{pmatrix}\quad&\text{if }x=\infty\end{cases}

be a matrix in $\operatorname{SL}_{2}(\mathbb{R})$ . Since $\sigma^{*}\eta_{x^{\prime},x}=\eta_{0,\infty}$ and $\sigma^{*}ds=ds$ , it is enough to show when $x=\infty$ and $x^{\prime}=0$ . On the geodesic $S_{0,\infty}=\{t\sqrt{-1}\mid t\in\mathbb{R}_{>0}\}$ , we have $ds=dt/t=\eta_{0,\infty}$ . ∎

2.2. Basic notation and properties for real quadratic numbers

For a real quadratic number $w$ , let $w^{\prime}$ be the non-trivial Galois conjugate of $w$ , $\operatorname{SL}_{2}(\mathbb{Z})_{w}$ be the stabilizer of $w$ in $\operatorname{SL}_{2}(\mathbb{Z})$ , and $\gamma_{w}=\begin{pmatrix}*&*\\ c&d\end{pmatrix}$ be the unique element of $\operatorname{SL}_{2}(\mathbb{Z})_{w}$ such that $\operatorname{SL}_{2}(\mathbb{Z})_{w}=\{\pm\gamma_{w}^{n}\mid n\in\mathbb{Z}\}$ and $\varepsilon_{w}:=cw+d>1$ . It holds $\gamma_{w}w^{\prime}=w^{\prime}$ . Thus, the geodesic $S_{w^{\prime},w}$ is invariant under $\gamma_{w}$ . It also holds $\gamma_{w}^{*}\eta_{w^{\prime},w}=\eta_{w^{\prime},w}$ by Lemma 2.2.

2.3. Basic notation and properties for continued fractions and words

For the convenience of the reader, we recall notations for even words prepared in Section 1.

Each real number can be represented by the unique continued fraction

(2.6)

[k_{1},k_{2},k_{3},\dots]:=k_{1}+\cfrac{1}{k_{2}+{\cfrac{1}{k_{3}+\cfrac{1}{\ddots}}}}

where $k_{1}$ is an integer and $k_{2},k_{3},\dots$ are positive integers. For positive integers $k_{1},\dots,k_{s}$ , we set a periodic continued fraction

(2.7)

[k_{1},\dots,k_{r},\overline{k_{r+1},\dots,k_{s}}]:=[k_{1},\dots,k_{r},k_{r+1},\dots,k_{s},k_{r+1},\dots,k_{s},k_{r+1},\dots,k_{s},\dots].

For a real number, being quadratic (resp. rational) is equivalent to having a periodic (resp. finite) continued fraction expansion ([Aig97, Theorem 1.17]).

For an even number $2r\in 2\mathbb{Z}_{\geq 0}$ and positive integers $k_{1},\dots,k_{2r}$ , we call a tuple $W=(k_{1},\dots,k_{2r})$ an even word. In the case when $r=0$ , we call it the empty word and denote it by $\emptyset$ . For even words $W_{1}=(k_{1}^{(1)},\dots,k_{2r_{1}}^{(1)}),\dots,W_{n}=(k_{1}^{(n)},\dots,k_{2r_{n}}^{(n)})$ , define their product

(2.8)

W_{1}\cdots W_{n}:=(k_{1}^{(1)},\dots,k_{2r_{1}}^{(1)},\dots,k_{1}^{(n)},\dots,k_{2r_{n}}^{(n)}).

For an infinite sequence of even words $W_{1}=(k_{1}^{(1)},\dots,k_{2r_{1}}^{(1)}),\dots,W_{n}=(k_{1}^{(n)},\dots,k_{2r_{n}}^{(n)}),\dots$ , define

(2.9)

[W_{1}\cdots W_{n}\cdots]:=[k_{1}^{(1)},\dots,k_{2r_{1}}^{(1)},\dots,k_{1}^{(n)},\dots,k_{2r_{n}}^{(n)},\dots].

For the empty word $\emptyset$ and a non-empty even word $W=(k_{1},\dots,k_{2r})$ , define

(2.10)	$\displaystyle\gamma_{\emptyset}$	$\displaystyle:=\begin{pmatrix}1&0\\ 0&1\end{pmatrix}\in\operatorname{SL}_{2}(\mathbb{Z}),$
(2.11)	$\displaystyle\gamma_{W}$	$\displaystyle:=\begin{pmatrix}k_{1}&1\\ 1&0\end{pmatrix}\begin{pmatrix}k_{2}&1\\ 1&0\end{pmatrix}\dots\begin{pmatrix}k_{2r}&1\\ 1&0\end{pmatrix}\in\operatorname{SL}_{2}(\mathbb{Z}),$
(2.12)	$\displaystyle\left\lvert W\right\rvert$	$\displaystyle:=r,$
(2.13)	$\displaystyle N(W)$	$\displaystyle:=\max\{n\in\mathbb{Z}_{>0}\mid\text{ there exists an even word }W_{1}\text{ such that }W=W_{1}^{n}\}.$

Moreover, for an even word $V=(l_{1},\dots,l_{2s})$ and a non-empty even word $W=(k_{1},\dots,k_{2r})$ , define

(2.14)

\displaystyle[V\overline{W}]

\displaystyle:=[l_{1},\dots,l_{2s},k_{1},\dots,k_{2r},k_{1},\dots,k_{2r},k_{1},\dots,k_{2r},\dots].

The matrix $\gamma_{W}$ has the following properties.

Lemma 2.4 ([Mur20, Lemma 2.3]).

The following properties hold.

(i)

For even words $W_{1},\dots,W_{n}$ , we have $\gamma_{W_{1}\cdots W_{n}}=\gamma_{W_{1}}\dots\gamma_{W_{n}}$ .
(ii)

Let $V$ , be an even word and $W\in\mathbb{Z}_{>0}^{\mathbb{N}}$ be an infinite word. Then, for an irrational number $x:=[W]$ , we have $\gamma_{V}x=[VW]$ . In particular, for a real quadratic number $v:=[\overline{V}]$ , we have $\gamma_{V}v=v$ .
(iii)

For a reduced real quadratic number $w$ , there exists the minimal even word $W$ such that $w=[\overline{W}]$ . Then we have $N(W)=1$ and $\gamma_{W}=\gamma_{w}$ . Generally, for a non-empty even word $W$ and a real quadratic number $w=[\overline{W}]$ , we have $\gamma_{W}=\gamma_{w}^{N(W)}$ .
(iv)

For a real quadratic number $w$ , there exist even words $V$ and $W$ such that $w=[V\overline{W}]$ . Then $\gamma_{V}\gamma_{W}\gamma_{V}^{-1}=\gamma_{w}^{N(W)}$ .

2.4. Kaneko’s val

Let $j\colon\mathbb{H}\to\mathbb{C}$ be the elliptic modular $j$ -function. For a real quadratic number $w$ , define

(2.15)

\widetilde{\operatorname{val}}(w):=\int_{z_{0}}^{\gamma_{w}z_{0}}j\eta_{w^{\prime},w},\quad\widetilde{1}(w):=\int_{z_{0}}^{\gamma_{w}z_{0}}\eta_{w^{\prime},w},\quad\operatorname{val}(w):=\frac{\widetilde{\operatorname{val}}(w)}{\widetilde{1}(w)},

where $z_{0}\in\mathbb{H}$ is any point.

The following lemma follows from Lemmas 2.2 and 2.3.

Lemma 2.5.

For a real quadratic number $w$ , any point $z_{0}\in S_{w^{\prime},w}$ and any positive integer $n$ , it holds

(2.16)		$\displaystyle n\widetilde{\operatorname{val}}(w)$	$\displaystyle=\int_{z_{0}}^{\gamma_{w}^{n}z_{0}}j\eta_{w^{\prime},w}=\int_{z_{0}}^{\gamma_{w}^{n}z_{0}}jds,\quad$
(2.17)		$\displaystyle n\widetilde{1}(w)$	$\displaystyle=\int_{z_{0}}^{\gamma_{w}^{n}z_{0}}\eta_{w^{\prime},w}=\int_{z_{0}}^{\gamma_{w}^{n}z_{0}}ds=d_{\mathrm{hyp}}(z_{0},\gamma_{w}^{n}z_{0}).$

The following lemma is fundamental.

Lemma 2.6 ([Mur20, Propositon 2.7]).

For a real quadratic number $w$ , it holds

(2.18)

\widetilde{1}(w)=2\log\varepsilon_{w}.

By the above lemmas, we obtain the following expression of $\operatorname{val}(w)$ , which are suitable to generalize for badly approximable numbers.

Proposition 2.7.

For a real quadratic number $w$ and any point $z_{0}\in S_{w^{\prime},w}$ , it holds

(2.19)

\displaystyle\operatorname{val}(w)

\displaystyle=\lim_{\begin{subarray}{c}z\in S_{z_{0},x}\\ z\to w\end{subarray}}\frac{1}{d_{\mathrm{hyp}}(z_{0},z)}\int_{S_{z_{0},z}}jds.

Proof.

For any positive number $d\in\mathbb{R}_{>0}$ , there exists the unique point $z_{d}\in S_{z_{0},w}$ such that $d_{\mathrm{hyp}}(z_{0},z_{d})=d$ . Let $l:=d_{\mathrm{hyp}}(z_{0},\gamma_{w}z_{0})=\widetilde{1}(w)$ . It suffices to show that

(2.20)

\lim_{n\to\infty}\frac{1}{nl+d}\int_{S_{z_{0},z_{nl+d}}}jds=\operatorname{val}(w)

for any number $0\leq d<l$ . Since $\gamma_{w}^{n}z_{d}\in S_{z_{0},w}$ and $d_{\mathrm{hyp}}(z_{d},\gamma_{w}^{n}z_{d})=nl$ by Lemma 2.5, we have $z_{nl+d}=\gamma_{w}^{n}z_{d}$ . Thus, we obtain

(2.21)	$\displaystyle\lim_{n\to\infty}\frac{1}{nl+d}\int_{S_{z_{0},z_{nl+d}}}jds$	$\displaystyle=\lim_{n\to\infty}\frac{1}{nl+d}\left(\int_{S_{z_{0},z_{d}}}+\int_{S_{z_{d},\gamma_{w}^{n}z_{d}}}\right)jds$
(2.22)		$\displaystyle=\lim_{n\to\infty}\frac{1}{nl+d}\left(\int_{S_{z_{0},z_{d}}}jds+n\widetilde{\operatorname{val}}(w)\right)$
(2.23)		$\displaystyle=\operatorname{val}(w)$

by Lemma 2.5. ∎

To generalize $\widetilde{\operatorname{val}}(w)$ , $\widetilde{1}(w)$ , and $\varepsilon_{w}$ for badly approximable numbers, we need some notation. For a real quadratic number $w=[V\overline{W}]$ , define

(2.24)

\widehat{\operatorname{val}}(w)=\frac{N(W)}{\left\lvert W\right\rvert}\widetilde{\operatorname{val}}(w),\quad\widehat{1}(w)=\frac{N(W)}{\left\lvert W\right\rvert}\widetilde{1}(w),\quad\widehat{\varepsilon}_{w}=\varepsilon_{w}^{N(W)/\left\lvert W\right\rvert}.

These values are independent of a choice of an even word $W$ and have the following expression.

Proposition 2.8.

Let $w$ be a real quadratic number. Let $r\geq 0$ and $s\geq 1$ be integers and $W_{1},\dots,W_{r+s}$ be non-empty even words such that $w=[W_{1}\cdots W_{r}\overline{W_{r+1}\cdots W_{r+s}}]$ . For an integer $n>r+s$ , let $W_{n}:=W_{r+l}$ where $0\leq l<s$ is an integer such that $n\equiv r+l\bmod s$ . For a positive integer $i$ , let $\gamma_{i}:=\gamma_{W_{i}}$ . Let $z_{0}\in\mathbb{H}$ be a point. Then, we have

(2.25)		$\displaystyle\widehat{\operatorname{val}}(w)$	$\displaystyle=\lim_{n\to\infty}\frac{1}{n}\int_{z_{0}}^{\gamma_{1}\cdots\gamma_{n}z_{0}}j\eta_{w^{\prime},w},$
(2.26)		$\displaystyle\widehat{1}(w)$	$\displaystyle=\lim_{n\to\infty}\frac{1}{n}\int_{z_{0}}^{\gamma_{1}\cdots\gamma_{n}z_{0}}\eta_{w^{\prime},w}.$

Proof.

Let $N:=N(W_{r+1}\cdots W_{r+s})$ . For a positive integer $n$ , let

(2.27)

a_{n}:=\int_{\gamma_{1}\cdots\gamma_{n-1}z_{0}}^{\gamma_{1}\cdots\gamma_{n}z_{0}}j\eta_{w^{\prime},w}.

Since

(2.28)

\frac{a_{1}+\dots+a_{n}}{n}=\frac{1}{n}\int_{z_{0}}^{\gamma_{1}\cdots\gamma_{n}z_{0}}j\eta_{w^{\prime},w},\quad\widehat{\operatorname{val}}(w)=\frac{N}{s}\widetilde{\operatorname{val}}(w),

it suffices to show that

(2.29)

\lim_{n\to\infty}\frac{a_{1}+\dots+a_{r+l+ns}}{r+l+ns}=\frac{N}{s}\widetilde{\operatorname{val}}(w)

for any integer $0\leq l<s$ .

Since $\gamma_{m+s}=\gamma_{m}$ for any integer $m>r+s$ , it holds

(2.30)

\displaystyle\gamma_{1}\cdots\gamma_{r+l+ns}

\displaystyle=\left(\gamma_{1}\cdots\gamma_{r+l}\gamma_{r+l+1}\cdots\gamma_{r+l+s}(\gamma_{1}\cdots\gamma_{r+l})^{-1}\right)^{n}\gamma_{1}\cdots\gamma_{r+l}=\gamma_{w}^{Nn}\gamma_{1}\cdots\gamma_{r+l}

by Lemma 2.4 (iv). Let $z^{\prime}_{0}:=\gamma_{1}\cdots\gamma_{r+l}z_{0}$ . Then, we have

(2.31)	$\displaystyle a_{r+l}+\dots+a_{r+l+ns}$	$\displaystyle=\int_{\gamma_{1}\cdots\gamma_{r+l}z_{0}}^{\gamma_{1}\cdots\gamma_{r+l+ns}z_{0}}j\eta_{w^{\prime},w}$
(2.32)		$\displaystyle=\int_{z^{\prime}_{0}}^{\gamma_{w}^{Nn}z^{\prime}_{0}}j\eta_{w^{\prime},w}$
(2.33)		$\displaystyle=Nn\widetilde{\operatorname{val}}(w)$

by Lemma 2.5. Thus, we obtain 2.29. The second equality is similarly proved. ∎

Remark 2.9.

In Proposition 2.8, the first finite term of a continued fraction does not contribute to the left-hand side. Thus, the right-hand side in Proposition 2.8 depends only on the period in the continued fraction.

3. Fundamental properties of the extended $\operatorname{val}$ function

With reference to Propositions 2.7 and 2.8, we now define $\operatorname{val}(x)$ for a badly approximable number $x$ as follows. Let $W_{1},W_{2},\dots$ be an infinite sequence of even words such that $x=[W_{1}W_{2}\cdots]$ and the set $\{W_{1},W_{2},\dots\}$ is finite. We remark that if $x<1$ then the first entry of $W_{1}$ is not positive. Let $c_{n}\in\mathbb{Z}_{>0}$ be the denominator of the rational number $[W_{1}\cdots W_{n}]$ and put

(3.1)

L:=\lim_{n\to\infty}\frac{\left\lvert W_{1}\right\rvert+\dots+\left\lvert W_{n}\right\rvert}{n}.

Take $x^{\prime}\in\mathbb{P}^{1}(\mathbb{R})\setminus\{x\}$ and $z_{0}\in\mathbb{H}$ . We define the limits $\operatorname{val}(x),\widehat{\operatorname{val}}(x),\widehat{1}(x),$ and $\widehat{\varepsilon}_{x}$ by

(3.2)	$\displaystyle\operatorname{val}(x)$	$\displaystyle:=\lim_{\begin{subarray}{c}z\in S_{z_{0},x}\\ z\to x\end{subarray}}\frac{1}{d_{\mathrm{hyp}}(z_{0},z)}\int_{S_{z_{0},z}}jds,$
(3.3)	$\displaystyle\widehat{\operatorname{val}}(x)$	$\displaystyle:=\lim_{n\to\infty}\frac{1}{\left\lvert W_{1}\right\rvert+\dots+\left\lvert W_{n}\right\rvert}\int_{z_{0}}^{\gamma_{W_{1}\cdots W_{n}}z_{0}}j\eta_{x^{\prime},x},$
(3.4)	$\displaystyle\widehat{1}(x)$	$\displaystyle:=\lim_{n\to\infty}\frac{1}{\left\lvert W_{1}\right\rvert+\dots+\left\lvert W_{n}\right\rvert}\int_{z_{0}}^{\gamma_{W_{1}\cdots W_{n}}z_{0}}\eta_{x^{\prime},x},$
(3.5)	$\displaystyle\widehat{\varepsilon}_{x}$	$\displaystyle:=\lim_{n\to\infty}c_{n}^{1/Ln}.$

Refer to caption — Figure 1. The paths for the definitions of $\operatorname{val}(x),\widehat{\operatorname{val}}(x)$ , and $\widehat{1}(x)$ in 3.2, 3.3, and 3.4

In the later sections, we will prove the following.

Theorem 3.1.

Let $x$ be a badly approximable number and $W_{1},W_{2},\dots$ be an infinite sequence of even words such that $x=[W_{1}W_{2}\cdots]$ and the set $\{W_{1},W_{2},\dots\}$ is finite. Take any points $x^{\prime}\in\mathbb{P}^{1}(\mathbb{R})\setminus\{x\}$ and $z_{0}\in\mathbb{H}$ . Then the following statements hold.

(i)

The limit values $\operatorname{val}(x),\widehat{\operatorname{val}}(x),\widehat{1}(x),$ and $\widehat{\varepsilon}_{x}$ defined in each of 3.2, 3.3, 3.4, and 3.5 are bounded. Moreover, if they converge then they are independent of $x^{\prime},z_{0}$ and $W_{1},W_{2},\dots$ . Further, they are $\operatorname{SL}_{2}(\mathbb{Z})$ -invariant.
(ii)

If $x$ is a real quadratic number, then $\operatorname{val}(x),\widehat{\operatorname{val}}(x),\widehat{1}(x),$ and $\widehat{\varepsilon}_{x}$ converge and coincide with the values defined in 1.1 and 2.24.

(iii)

If $\widehat{\operatorname{val}}(x)$ and $\widehat{1}(x)$ converge, then for any point $x^{\prime}\in\mathbb{P}^{1}(\mathbb{R})\setminus\{x\}$ and a sequence $\{x_{n}\}_{n=1}^{\infty}$ of real numbers with $z_{n}:=\gamma_{W_{1}\cdots W_{n}}(x_{n}+\sqrt{-1})\in S_{\infty,x},z_{n}\to x$ , we have

(3.6)		$\displaystyle\widehat{\operatorname{val}}(x)$	$\displaystyle=\lim_{n\to\infty}\frac{1}{\left\lvert W_{1}\right\rvert+\dots+\left\lvert W_{n}\right\rvert}\int_{z_{0}}^{z_{n}}j\eta_{x^{\prime},x},$
(3.7)		$\displaystyle\widehat{1}(x)$	$\displaystyle=\lim_{n\to\infty}\frac{1}{\left\lvert W_{1}\right\rvert+\dots+\left\lvert W_{n}\right\rvert}\int_{z_{0}}^{z_{n}}\eta_{x^{\prime},x}.$

Here we define $1/(z-\infty):=0$ if $x^{\prime}=\infty$ .

(iv)

The limit $\widehat{\varepsilon}_{x}$ converges if and only if $\widehat{1}(x)$ converges. Moreover, we have $\widehat{1}(x)=2\log\widehat{\varepsilon}_{x}$ .
(v)

It holds $\widehat{\varepsilon}_{x}\geq(3+\sqrt{5})/2$ . In particular, $\widehat{1}(x)>0$ .
(vi)

If $\widehat{\operatorname{val}}(x)$ and $\widehat{1}(x)$ converge, then $\operatorname{val}(x)$ converges and we have $\operatorname{val}(x)=\widehat{\operatorname{val}}(x)/\widehat{1}(x)$ .

Here we remark that Theorem 3.1 (vi) follows from Theorem 3.1 (iv) and (v).

4. The extended $\operatorname{val}$ function as the limit value along a geodesic

In this section, we prove Theorem 3.1 (i) and (ii) for $\operatorname{val}(x)$ . We start with the following properties of badly approximable numbers. Let $\pi\colon\mathbb{H}\to\operatorname{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}$ be the natural projection.

Proposition 4.1 ([Dal11, Chapter VII, Theorem 3.4], [Dal11, Chapter VII, Lemma 3.5]).

For an irrational number $x\in\mathbb{R}\setminus\mathbb{Q}$ , the following statements are equivalent.

(i)

$x$ is badly approximable;
(ii)

$L(x):=\sup\left\{L>0\mathrel{}\middle|\mathrel{}\exists p_{n},q_{n}\in\mathbb{Z}\text{ s.t. }(p_{n},q_{n})=1,q_{n}\to\infty,\left\lvert x-p_{n}/q_{n}\right\rvert\leq 1/Lq_{n}^{2}\right\}<\infty$ ;
(iii)

$h(x):=\sup\left\{t>0\mid\exists z_{n}\in S_{z_{0},x}\text{ s.t. }z_{n}\to x,\pi(z_{n})\in\pi(\{z\in\mathbb{H}\mid\operatorname{Im}(z)=t\})\right\}<\infty$ ;
(iv)

For any point $z_{0}\in\mathbb{H}$ , the closure $\overline{\pi(S_{z_{0},x})}\subset\operatorname{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}$ is compact.

The constant $L(x)$ is called the Lagrange number of $x$ . Usually, badly approximable numbers are defined as irrational numbers with finite Lagrange numbers.

The following is a key fact in this section.

Theorem 4.2 (Theorem 3.1 (i) and (ii) for $\operatorname{val}(x)$ ).

Let $x$ be a badly approximable number. Take any point $z_{0}\in\mathbb{H}$ . Then the following statements hold.

(i)

The limit value

(4.1)

\operatorname{val}(x):=\lim_{\begin{subarray}{c}z\in S_{z_{0},x}\\ z\to x\end{subarray}}\frac{1}{d_{\mathrm{hyp}}(z_{0},z)}\int_{S_{z_{0},z}}jds

is bounded.

(ii)

For any matrix $\gamma\in\operatorname{SL}_{2}(\mathbb{Z})$ , we have $\operatorname{val}(\gamma x)=\operatorname{val}(x)$ .
(iii)

If the limit $\operatorname{val}(x)$ converges, then it is independent of $z_{0}$ .
(iv)

For a real quadratic number $x$ , $\operatorname{val}(x)$ converges and coincides with the value defined in 1.1.

Proof.

To begin with, we remark that (iv) follows from Propositions 2.7 and (iii).

We will prove (i). Since $x$ is badly approximable, the closure $\overline{\pi(S_{x,z_{0}})}\subset\operatorname{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}$ is compact by Proposition 4.1. Thus, we can choose $K>0$ such that $\left\lvert j(z)\right\rvert\leq K$ on $S_{z_{0},x}$ . Since

(4.2)

\displaystyle\left\lvert\frac{1}{d_{\mathrm{hyp}}(z_{0},z)}\int_{S_{z_{0},z}}jds\right\rvert=\frac{1}{d_{\mathrm{hyp}}(z_{0},z)}\int_{S_{z_{0},z}}Kds=K,

the limit is bounded.

The second claim (ii) follows from the fact that $j(z)$ is $\operatorname{SL}_{2}(\mathbb{Z})$ -invariant.

As for the third claim (iii), for each point $z_{0}\in\mathbb{H}$ , we consider

(4.3)

\operatorname{val}(x,z_{0}):=\lim_{\begin{subarray}{c}z\in S_{z_{0},x}\\ z\to x\end{subarray}}\frac{1}{d_{\mathrm{hyp}}(z_{0},z)}\int_{S_{z_{0},z}}jds.

We will show that this value is independent of $z_{0}$ in two steps.

Step 1. We will show that $\operatorname{val}(x,z_{0})=\operatorname{val}(x,z_{1})$ for any point $z_{1}\in S_{z_{0},x}$ . For $i\in\{0,1\}$ , let

(4.4)

a_{i}(z):=\int_{S_{z_{i},z}}jds,\quad b_{i}(z):=\int_{S_{z_{i},z}}ds.

Clearly,

(4.5)

\lim_{\begin{subarray}{c}z\in S_{z_{0},x}\\ z\to x\end{subarray}}b_{i}(z)=\infty.

Since

(4.6)

a_{0}(z)-a_{1}(z)=\int_{S_{z_{0},z_{1}}}jds,\quad b_{0}(z)-b_{1}(z)=\int_{S_{z_{0},z_{1}}}ds

is bounded, we have

(4.7)

\operatorname{val}(x,z_{0})=\lim_{\begin{subarray}{c}z\in S_{z_{0},x}\\ z\to x\end{subarray}}\frac{a_{0}(z)}{b_{0}(z)}=\lim_{\begin{subarray}{c}z\in S_{z_{0},x}\\ z\to x\end{subarray}}\frac{a_{1}(z)+O(1)}{b_{1}(z)+O(1)}=\lim_{\begin{subarray}{c}z\in S_{z_{0},x}\\ z\to x\end{subarray}}\frac{a_{1}(z)}{b_{1}(z)}=\operatorname{val}(x,z_{1}).

Step 2. We will show that $\operatorname{val}(x,z_{0})=\operatorname{val}(x,z_{1})$ for any points $z_{0},z_{1}\in\mathbb{H}$ . This part is due to Matsusaka. Take a matrix $\sigma\in\operatorname{SL}_{2}(\mathbb{R})$ such that $\sigma^{-1}x=\infty$ . Let $v_{0}$ be any positive number. By Step 1, we can replace $z_{i}$ by $\operatorname{Re}(z_{i})+v_{0}\sqrt{-1}$ . Thus, we may assume $v_{0}=\operatorname{Im}(\sigma^{-1}z_{0})=\operatorname{Im}(\sigma^{-1}z_{1})$ . Let $u_{i}:=\operatorname{Re}(\sigma^{-1}z_{i})$ for $i\in\{0,1\}$ . Then $\sigma^{-1}S_{z_{i},x}$ is a subgeodesic of $S_{u_{i},\infty}$ . Since $ds=y^{-1}\sqrt{dx^{2}+dy^{2}}=y^{-1}dy$ on $S_{u_{i},\infty}$ , we have

(4.8)		$\displaystyle\operatorname{val}(x,z_{i})$	$\displaystyle=\lim_{\begin{subarray}{c}z\in S_{z_{i},x}\\ z\to x\end{subarray}}\frac{1}{d_{\mathrm{hyp}}(z_{i},z)}\int_{S_{z_{i},z}}jds$
(4.9)			$\displaystyle=\lim_{v\to\infty}\frac{1}{d_{\mathrm{hyp}}(\sigma(u_{i}+v\sqrt{-1}),\sigma(u_{i}+v_{0}\sqrt{-1}))}\int_{v_{0}}^{v}j(\sigma(u_{i}+y\sqrt{-1}))\frac{dy}{y}.$

Since

(4.10)

\displaystyle d_{\mathrm{hyp}}(\sigma(u_{i}+v\sqrt{-1}),\sigma(u_{i}+v_{0}\sqrt{-1}))=d_{\mathrm{hyp}}(v\sqrt{-1},v_{0}\sqrt{-1})=\int_{v_{0}}^{v}\frac{dy}{y}=\log\frac{v}{v_{0}},

we obtain

(4.11)

\operatorname{val}(x,z_{0})-\operatorname{val}(x,z_{1})=\lim_{v\to\infty}\frac{1}{\log v/v_{0}}\int_{v_{0}}^{v}\left(j(\sigma(u_{0}+y\sqrt{-1}))-j(\sigma(u_{1}+y\sqrt{-1}))\right)\frac{dy}{y}.

For two points $z,z^{\prime}\in\mathbb{H}$ , the hyperbolic distances on $\mathbb{H}$ and $\operatorname{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}$ are respectively defined by

(4.12)

d_{\mathrm{hyp}}(z,z^{\prime}):=\operatorname{length}(S_{z,z^{\prime}}),\quad d_{\operatorname{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}}(\pi(z),\pi(z^{\prime})):=\min\left\{d_{\mathrm{hyp}}(\gamma z,z^{\prime})\mid\gamma\in\operatorname{SL}_{2}(\mathbb{Z})\right\}.

They define the hyperbolic distances on $\mathbb{H}$ and $\operatorname{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}$ respectively. By Lemma 2.1, we have

(4.13)

\cosh d_{\mathrm{hyp}}(\sigma(u_{0}+y\sqrt{-1}),\sigma(u_{1}+y\sqrt{-1}))=\cosh d_{\mathrm{hyp}}(u_{0}+y\sqrt{-1},u_{1}+y\sqrt{-1})=1+\frac{\left\lvert u_{0}-u_{1}\right\rvert^{2}}{2y^{2}}.

Thus, we have

(4.14)

\lim_{y\to\infty}d_{\mathrm{hyp}}(\sigma(u_{0}+y\sqrt{-1}),\sigma(u_{1}+y\sqrt{-1}))=0.

Since $x$ is badly approximable, $\overline{\pi(S_{x,z_{0}})}\cup\overline{\pi(S_{x,z_{1}})}\subset\operatorname{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}$ is compact by Proposition 4.1. Thus, $j(z)$ is continuous on $\overline{\pi(S_{x,z_{0}})}\cup\overline{\pi(S_{x,z_{1}})}$ with respect to the metric $d_{\operatorname{SL}_{2}(\mathbb{Z})\backslash\mathbb{H}}$ . Hence for any $\varepsilon>0$ , there exists $v_{0}>0$ such that

(4.15)

\left\lvert j(\sigma(u_{0}+y\sqrt{-1}))-j(\sigma(u_{1}+y\sqrt{-1}))\right\rvert<\varepsilon

for any $y>v_{0}$ . Then we have

(4.16)

\left\lvert\operatorname{val}(x,z_{0})-\operatorname{val}(x,z_{1})\right\rvert\leq\lim_{v\to\infty}\frac{1}{\log v/v_{0}}\int_{v_{0}}^{v}\varepsilon\frac{dy}{y}=\varepsilon,

that is, $\operatorname{val}(x,z_{0})=\operatorname{val}(x,z_{1})$ . ∎

5. The extended $\operatorname{val}$ function as the limit value along a continued fraction

In this section, we prove Theorem 3.1 (i), (ii), and (iii) for $\widehat{\operatorname{val}}(x)$ and $\widehat{1}(x)$ .

To prove Theorem 3.1 (ii), we prepare several lemmas.

Lemma 5.1.

For a badly approximable number $x=[k_{1},k_{2},\dots]$ , let $\gamma_{n}:=\gamma_{(k_{1},\dots,k_{2n})}$ . Then for any point $x^{\prime}\in\mathbb{P}^{1}(\mathbb{R})$ , the set $\{\gamma_{n}^{-1}x^{\prime}\mid n\in\mathbb{Z}_{>0}\}$ is bounded.

Proof.

Let $h(x)$ be the number defined in Proposition 4.1. By Proposition 4.1, we have

(5.1)

\gamma_{n}^{-1}S_{x^{\prime},x}\cap\{z\in\mathbb{H}\mid\operatorname{Im}(z)\geq h(x)+1\}=\emptyset

for all sufficiently large $n$ . Thus, the radius of $\gamma_{n}^{-1}S_{x^{\prime},x}$ is less than $h(x)+1$ . Since coefficients of the continued fraction expansion of $x$ are bounded, $\{\gamma_{n}^{-1}x\mid n\in\mathbb{Z}_{>0}\}$ is bounded. Thus, $\{\gamma_{n}^{-1}x^{\prime}\mid n\in\mathbb{Z}_{>0}\}$ is also bounded. ∎

Lemma 5.2.

For an even word $W$ , a point $z_{0}\in\mathbb{H}$ , and sequences $\{x_{n}\},\{x^{\prime}_{n}\},\{y_{n}\},\{y^{\prime}_{n}\}$ , we have

(5.2)

\int_{z_{0}}^{\gamma_{W}z_{0}}j\left(\eta_{x_{n}^{\prime},x_{n}}-\eta_{y_{n}^{\prime},y_{n}}\right)=O(\left\lvert x_{n}-y_{n}\right\rvert+\left\lvert x^{\prime}_{n}-y^{\prime}_{n}\right\rvert).

Proof.

Since

(5.3)

\left\lvert\dfrac{1}{z-x_{n}}-\dfrac{1}{z-y_{n}}\right\rvert\leq\frac{\left\lvert x_{n}-y_{n}\right\rvert}{\min\{\operatorname{Im}(z_{0}),\operatorname{Im}(\gamma_{W}z_{0})\}}

on the contour $\{tz_{0}+(1-t)\gamma_{W}z_{0}\mid 0\leq t\leq 1\}$ , it holds

(5.4)

\left\lvert\int_{z_{0}}^{\gamma_{W}z_{0}}j(z)\left(\dfrac{1}{z-x_{n}}-\dfrac{1}{z-y_{n}}\right)\right\rvert\leq\frac{\left\lvert x_{n}-y_{n}\right\rvert}{\min\{\operatorname{Im}(z_{0}),\operatorname{Im}(\gamma_{W}z_{0})\}}\left\lvert\int_{z_{0}}^{\gamma_{W}z_{0}}j(z)dz\right\rvert=O(\left\lvert x_{n}-y_{n}\right\rvert).

Similar argument shows

(5.5)

\left\lvert\int_{z_{0}}^{\gamma_{W}z_{0}}j(z)\left(\dfrac{1}{z-x_{n}^{\prime}}-\dfrac{1}{z-y_{n}^{\prime}}\right)\right\rvert=O(\left\lvert x_{n}^{\prime}-y_{n}^{\prime}\right\rvert).

∎

A key result in this section is the following.

Theorem 5.3 (Theorem 3.1 (i) for $\widehat{\operatorname{val}}(x)$ and $\widehat{1}(x)$ ).

Let $x$ be a badly approximable number and $W_{1},W_{2},\dots$ be an infinite sequence of even words such that $x=[W_{1}W_{2}\cdots]$ and the set $\{W_{1},W_{2},\dots\}$ is finite. Take a point $z_{0}\in\mathbb{H}$ . Then the following statements hold.

(i)

The sequences defining the limits

(5.6)		$\displaystyle\widehat{\operatorname{val}}(x)$	$\displaystyle:=\lim_{n\to\infty}\frac{1}{\left\lvert W_{1}\right\rvert+\dots+\left\lvert W_{n}\right\rvert}\int_{z_{0}}^{\gamma_{W_{1}\cdots W_{n}}z_{0}}j\eta_{\infty,x},$
(5.7)		$\displaystyle\widehat{1}(x)$	$\displaystyle:=\lim_{n\to\infty}\frac{1}{\left\lvert W_{1}\right\rvert+\dots+\left\lvert W_{n}\right\rvert}\int_{z_{0}}^{\gamma_{W_{1}\cdots W_{n}}z_{0}}\eta_{\infty,x}$

is bounded.

(ii)

The limits $\widehat{\operatorname{val}}(x)$ and $\widehat{1}(x)$ are independent of $z_{0}$ .
(iii)

The limits $\widehat{\operatorname{val}}(x)$ and $\widehat{1}(x)$ are independent of $\{W_{n}\mid n\geq 1\}$ .
(iv)

The limits $\widehat{\operatorname{val}}(x)$ and $\widehat{1}(x)$ are $\operatorname{SL}_{2}(\mathbb{Z})$ -invariant.

Proof.

For the first claim (i), let

(5.8)	$\displaystyle L$	$\displaystyle:=\lim_{n\to\infty}\frac{\left\lvert W_{1}\right\rvert+\dots+\left\lvert W_{n}\right\rvert}{2n},$
(5.9)	$\displaystyle a_{n}(z_{0})$	$\displaystyle:=\int_{\gamma_{W_{1}\cdots W_{n-1}z_{0}}}^{\gamma_{W_{1}\cdots W_{n}}z_{0}}j\eta_{\infty,x},$
(5.10)	$\displaystyle K$	$\displaystyle:=\max\left\{\left\lvert j(z)\right\rvert\mathrel{}\middle\|\mathrel{}\left\lvert\operatorname{Re}(z)\right\rvert\leq 1/2,\left\lvert z\right\rvert\leq 1,\operatorname{Im}(z)\leq h(x)\right\},$
(5.11)	$\displaystyle R$	$\displaystyle:=\min\{\operatorname{Im}(z_{0}),\operatorname{Im}(\gamma_{W_{n}}z_{0})\mid n\in\mathbb{Z}_{>0}\},$
(5.12)	$\displaystyle M$	$\displaystyle:=\sup\{\gamma_{W_{1}\cdots W_{n}}^{-1}x,\gamma_{W_{1}\cdots W_{n}}^{-1}\infty\mid n\in\mathbb{Z}_{>0}\}.$

Here $K$ and $R$ are positive real numbers and $L$ converges since the set $\{W_{1},W_{2},\dots\}$ is finite. Since $x$ is badly approximable, $M$ is a positive real number by Lemma 5.1. We can write

(5.13)

\widehat{\operatorname{val}}(x)=\frac{1}{L}\lim_{n\to\infty}\frac{a_{1}(z_{0})+\dots+a_{n}(z_{0})}{n}.

For any positive integer $n_{0}$ , we have

(5.14)	$\displaystyle\left\lvert a_{1}(z_{0})+\dots+a_{n}(z_{0})\right\rvert$	$\displaystyle\leq\int_{z_{0}}^{\gamma_{W_{1}\cdots W_{n}}z_{0}}\left\lvert j\eta_{\infty,x}\right\rvert$
(5.15)		$\displaystyle=\int_{z_{0}}^{\gamma_{W_{1}\cdots W_{n}}z_{0}}\left\lvert j(z)\frac{-1}{z-x}\right\rvert\left\lvert dz\right\rvert$
(5.16)		$\displaystyle\leq\frac{2MK}{R^{2}}n$

and thus the sequence $\{(a_{1}(z_{0})+\dots+a_{n}(z_{0}))/n\}_{n\geq 1}$ is bounded.

For (ii), pick any other point $z_{0}^{\prime}\in\mathbb{H}$ . We have

(5.17)	$\displaystyle\left(a_{1}(z_{0})+\dots+a_{n}(z_{0})\right)-\left(a_{1}(z^{\prime}_{0})+\dots+a_{n}(z^{\prime}_{0})\right)$	$\displaystyle=\left(\int_{\gamma_{W_{1}\cdots W_{n}}z_{0}^{\prime}}^{\gamma_{W_{1}\cdots W_{n}}z_{0}}-\int_{z_{0}^{\prime}}^{z_{0}}\right)j\eta_{\infty,x}$
(5.18)		$\displaystyle=\int_{z_{0}^{\prime}}^{z_{0}}j\left(\gamma_{W_{1}\cdots W_{n}}^{*}\eta_{\infty,x}-\eta_{\infty,x}\right)$
(5.19)		$\displaystyle=\int_{z_{0}^{\prime}}^{z_{0}}j\left(\eta_{\gamma_{W_{1}\cdots W_{n}}^{-1}\infty,\gamma_{W_{1}\cdots W_{n}}^{-1}x}-\eta_{\infty,x}\right)$

by Lemma 2.2. Since the most right-hand side is bounded by Lemmas 5.1 and 5.2, we have

(5.20)

\lim_{n\to\infty}\frac{a_{1}(z_{0})+\dots+a_{n}(z_{0})}{n}=\lim_{n\to\infty}\frac{a_{1}(z^{\prime}_{0})+\dots+a_{n}(z^{\prime}_{0})}{n}.

As for (iii), let $x=[k_{1},k_{2},\dots],V_{n}:=(k_{2n-1},k_{2n})$ , and $L_{n}:=\left\lvert W_{1}\right\rvert+\dots+\left\lvert W_{n}\right\rvert$ . Since $\gamma_{W_{1}\cdots W_{n}}=\gamma_{V_{1}\cdots V_{L_{n}}}$ , we have

(5.21)

\frac{1}{L_{n}}\int_{z_{0}}^{\gamma_{W_{1}\cdots W_{n}}z_{0}}=\frac{1}{L_{n}}\int_{z_{0}}^{\gamma_{V_{1}\cdots V_{L_{n}}}z_{0}}

and thus the values $\widehat{\operatorname{val}}(x)$ and $\widehat{1}(x)$ are independent of $\{W_{n}\mid n\geq 1\}$ .

Finally, we prove (iv). Since $\operatorname{SL}_{2}(\mathbb{Z})$ -equivalent real numbers have the same continued fraction expansions except for the first few terms, it suffices to show $\widetilde{f}(x)=\widetilde{f}(\gamma^{-1}x)$ in the case when $x=[W_{1}W_{2}\dots],\gamma=\gamma_{W_{1}}\dots\gamma_{W_{k}}$ . This follows from the fact that

(5.22)

\int_{\gamma_{W_{1}}\dots\gamma_{W_{m}}z_{0}}^{\gamma_{W_{1}}\dots\gamma_{W_{m+n+k}}z_{0}}j\eta_{\infty,x}=\int_{\gamma_{W_{k+1}}\dots\gamma_{W_{m}}z_{0}}^{\gamma_{W_{k+1}}\dots\gamma_{W_{m+n+k}}z_{0}}j\eta_{\gamma^{-1}\infty,\gamma^{-1}x}

for $m>k$ is bounded by Lemmas 5.1 and 5.2. ∎

In Theorem 5.3 (i), we consider the differential form $\eta_{\infty,x}$ . In fact, we can replace it by $\eta_{x^{\prime},x}$ for any point $x^{\prime}\in\mathbb{P}^{1}(\mathbb{R})\setminus\{x\}$ . To prove it, we prepare the following lemma.

Lemma 5.4.

In the setting of Theorem 5.3, let $\{x^{\prime}_{n}\}_{n=1}^{\infty}\subset\mathbb{R}$ be a bounded sequence such that $x$ is not an accumulation point. Then the sequence

(5.23)

\int_{z_{0}}^{\gamma_{W_{1}\cdots W_{n}}z_{0}}j(z)\frac{dz}{z-x^{\prime}_{n}}

is bounded.

Proof.

Let

(5.24)

K:=\max\left\{\left\lvert j(z)\right\rvert\mathrel{}\middle|\mathrel{}\left\lvert\operatorname{Re}(z)\right\rvert\leq 1/2,\left\lvert z\right\rvert\leq 1,\operatorname{Im}(z)\leq h(x)\right\}.

Then

(5.25)		$\displaystyle\left\lvert\int_{z_{0}}^{\gamma_{W_{1}\cdots W_{n}}z_{0}}j(z)\frac{dz}{z-x^{\prime}_{n}}\right\rvert\leq K\int_{S_{z_{0},\gamma_{W_{1}\cdots W_{n}}z_{0}}}\frac{\left\lvert dz\right\rvert}{\left\lvert z-x^{\prime}_{n}\right\rvert}$
(5.26)		$\displaystyle\leq K\int_{S_{z_{0},x}}\frac{\left\lvert dz\right\rvert}{\left\lvert z-x^{\prime}_{n}\right\rvert}.$

∎

By Lemma 5.4, we can reformulate the definition of $\widehat{\operatorname{val}}(x)$ in Theorem 5.3 (i) as follows.

Proposition 5.5.

In the setting of Theorem 5.3, let $\{x^{\prime}_{n}\}_{n=1}^{\infty}\subset\mathbb{R}$ be a bounded sequence such that $x$ is not an accumulation point. Then

(5.27)

\widehat{\operatorname{val}}(x)=\lim_{n\to\infty}\frac{1}{\left\lvert W_{1}\right\rvert+\dots+\left\lvert W_{n}\right\rvert}\int_{z_{0}}^{\gamma_{W_{1}\cdots W_{n}}z_{0}}j\eta_{x^{\prime}_{n},x}

holds.

Finally, we prove Theorem 3.1 (iii).

Theorem 5.6 (Theorem 3.1 (iii)).

In the setting of Theorem 5.3, for any point $x^{\prime}\in\mathbb{P}^{1}(\mathbb{R})\setminus\{x\}$ and a sequence $\{x_{n}\}_{n=1}^{\infty}$ of real numbers such that $z_{n}:=\gamma_{W_{1}\cdots W_{n}}(x_{n}+\sqrt{-1})\in S_{\infty,x}$ and $z_{n}\to x$ , we have

(5.28)		$\displaystyle\widehat{\operatorname{val}}(x)$	$\displaystyle=\lim_{n\to\infty}\frac{1}{\left\lvert W_{1}\right\rvert+\dots+\left\lvert W_{n}\right\rvert}\int_{z_{0}}^{z_{n}}j\eta_{x^{\prime},x},$
(5.29)		$\displaystyle\widehat{1}(x)$	$\displaystyle=\lim_{n\to\infty}\frac{1}{\left\lvert W_{1}\right\rvert+\dots+\left\lvert W_{n}\right\rvert}\int_{z_{0}}^{z_{n}}\eta_{x^{\prime},x}.$

Proof.

We prove only the first equality. Let $\gamma_{n}:=\gamma_{W_{1}\cdots W_{n}}$ . By Proposition 5.5, we have

(5.30)

\widehat{\operatorname{val}}(x)=\lim_{n\to\infty}\frac{1}{\left\lvert W_{1}\right\rvert+\dots+\left\lvert W_{n}\right\rvert}\int_{\sqrt{-1}}^{\gamma_{n}\sqrt{-1}}j\eta_{x^{\prime},x}.

Take a point $z_{0}\in S_{x^{\prime},x}$ such that $\pi(z_{0})\in\pi(\{z\in\mathbb{H}\mid 1\leq\operatorname{Im}(z)\leq h(x)\})$ . Then we have

(5.31)

\widehat{\operatorname{val}}(x)=\lim_{n\to\infty}\frac{1}{\left\lvert W_{1}\right\rvert+\dots+\left\lvert W_{n}\right\rvert}\int_{z_{0}}^{\gamma_{n}\sqrt{-1}}j\eta_{x^{\prime},x}.

Since $x_{n}+\sqrt{-1}\in\gamma_{n}^{-1}(S_{x^{\prime},x})$ , we have $\left\lvert x_{n}\right\rvert\leq\max\{\left\lvert\gamma_{n}^{-1}x\right\rvert,\left\lvert\gamma_{n}^{-1}x^{\prime}\right\rvert\}$ . Thus, $\left\lvert x_{n}\right\rvert$ is bounded by Lemma 5.1. Since

(5.32)

\left\lvert d(z_{n},z_{0})-d(\gamma_{n}\sqrt{-1},z_{0})\right\rvert\leq d(z_{n},\gamma_{n}\sqrt{-1})\leq d(x_{n}+\sqrt{-1},\sqrt{-1})=1+\frac{x_{n}^{2}}{2}

by the triangle inequality and Lemma 2.1 (ii), $d(z_{n},z_{0})-d(\gamma_{n}\sqrt{-1},z_{0})$ is bounded. Let

(5.33)

K:=\max\left\{\left\lvert j(z)\right\rvert\mathrel{}\middle|\mathrel{}\left\lvert\operatorname{Re}(z)\right\rvert\leq 1/2,\left\lvert z\right\rvert\leq 1,\operatorname{Im}(z)\leq h(x)\right\}.

Then

(5.34)	$\displaystyle\left(\int_{z_{0}}^{z_{n}}-\int_{z_{0}}^{\gamma_{n}\sqrt{-1}}\right)j\eta_{x^{\prime},x}$	$\displaystyle=\int_{\sqrt{-1}}^{x_{n}+\sqrt{-1}}j\eta_{\gamma_{n}^{-1}x^{\prime},\gamma_{n}^{-1}x}$
(5.35)		$\displaystyle\leq K\int_{\sqrt{-1}}^{x_{n}+\sqrt{-1}}\frac{\left\lvert\gamma_{n}^{-1}x-\gamma_{n}^{-1}x^{\prime}\right\rvert}{(\min\{\operatorname{Im}(x_{n}+\sqrt{-1}),\operatorname{Im}(\sqrt{-1})\})^{2}}dz$
(5.36)		$\displaystyle\leq K\left\lvert\gamma_{n}^{-1}x-\gamma_{n}^{-1}x^{\prime}\right\rvert x_{n}$

is bounded by Lemma 5.1. Thus, we obtain

(5.37)

\widehat{\operatorname{val}}(x)=\lim_{n\to\infty}\frac{1}{\left\lvert W_{1}\right\rvert+\dots+\left\lvert W_{n}\right\rvert}\int_{z_{0}}^{z_{n}}j\eta_{x^{\prime},x}.

∎

6. Elementary units for badly approximable numbers

In this section, we consider an analog of elementary units for badly approximable numbers and prove Theorem 3.1 (iv) and (v).

Theorem 6.1 (Theorem 3.1 (iv) and (v)).

Let $x$ be a badly approximable number and $W_{1},W_{2},\dots$ be an infinite sequence of even words such that $x=[W_{1}W_{2}\cdots]$ and the set $\{W_{1},W_{2},\dots\}$ is finite. Let $c_{n}\in\mathbb{Z}_{>0}$ be the denominator of a rational number $[W_{1}\cdots W_{n}]$ and

(6.1)

L:=\lim_{n\to\infty}\frac{\left\lvert W_{1}\right\rvert+\dots+\left\lvert W_{n}\right\rvert}{n}.

Then the limit

(6.2)

\varepsilon_{x}:=\lim_{n\to\infty}c_{n}^{1/Ln}

converges if and only if $\widehat{1}(x)$ converges. In that case, it holds $\widehat{1}(x)=2\log\varepsilon_{x}$ .

Here we call $\varepsilon_{x}$ the elementary units for each badly approximable number $x$ . Before giving a proof, we state a remark.

Remark 6.2.

In the situation in Theorem 6.1, we have $\varepsilon(W_{1},W_{2},\dots)\geq(3+\sqrt{5})/2$ .

Proof of Theorem 6.1.

Let

(6.3)

\gamma_{n}:=\gamma_{W_{1}\cdots W_{n}}=\begin{pmatrix}*&*\\ c_{n}&d_{n}\end{pmatrix}.

Here $c_{n}$ is a denominator of the rational number $\gamma_{n}(\infty)=[W_{1}\cdots W_{n}]$ . Pick any point $z_{0}\in\mathbb{H}$ . Then we have

(6.4)

\displaystyle\widetilde{1}(x)=\lim_{n\to\infty}\frac{1}{n}\int_{z_{0}}^{\gamma_{n}z_{0}}\frac{-1}{z-x}dz=\lim_{n\to\infty}\frac{1}{n}\biggl{[}-\log(z-x)\biggr{]}_{z_{0}}^{\gamma_{n}z_{0}}.

Since $\gamma_{n}z_{0}$ converges to $x$ as $n\to\infty$ and the argument of $\log$ is bounded, this limit value is equal to

(6.5)

\displaystyle\lim_{n\to\infty}\frac{-1}{n}\log\left\lvert\gamma_{n}z_{0}-x\right\rvert=\lim_{n\to\infty}\frac{-1}{n}\log\left\lvert\frac{z_{0}-\gamma_{n}^{-1}x}{(c_{n}z_{0}+d_{n})(c_{n}\gamma_{n}^{-1}x+d_{n})}\right\rvert.

This is equal to

(6.6)

\lim_{n\to\infty}\frac{1}{n}\log\left\lvert(c_{n}z_{0}+d_{n})(c_{n}\gamma_{n}^{-1}x+d_{n})\right\rvert

since $x$ is badly approximable and $\{\gamma_{n}^{-1}x\mid n\in\mathbb{Z}_{>0}\}$ is bounded by Lemma 5.1. Since $\gamma_{n}^{-1}\infty=-d_{n}/c_{n}$ is bounded by Lemma 5.1,

(6.7)

\left\{\frac{c_{n}z+d_{n}}{c_{n}\gamma_{n}^{-1}x+d_{n}}\right\}_{n\in\mathbb{Z}_{>0}}

is bounded for any point $z\in\mathbb{H}\cup\mathbb{R}$ . Thus, we obtain

(6.8)

\widehat{1}(x)=\lim_{n\to\infty}\frac{2}{n}\log\left\lvert c_{n}z+d_{n}\right\rvert.

By substituting $z=0$ , we have

(6.9)

\widehat{1}(x)=\lim_{n\to\infty}\frac{2}{n}\log d_{n}.

Since $d_{n}/c_{n}$ is bounded, we obtain

(6.10)

\widehat{1}(x)=\lim_{n\to\infty}\frac{2}{n}\log c_{n}=2\log\varepsilon_{x}.

∎

Example 6.3.

For the case when $x=\phi:=(1+\sqrt{5})/2=[1,1,\dots]$ , since

(6.11)

[\underbrace{1,\dots,1}_{2n}]=\frac{F_{2n+1}}{F_{2n}}

where $F_{n}$ denotes the $n$ -th Fibonacci number, we have

(6.12)

\displaystyle\varepsilon_{\phi}

\displaystyle=\lim_{n\to\infty}\sqrt[n]{F_{2n}}=\lim_{n\to\infty}\sqrt[n]{\frac{1}{\sqrt{5}}\left(\phi^{2n}-(-\phi)^{2n}\right)}=\phi^{2}=\frac{3+\sqrt{5}}{2}.

7. An explicit computation of values of extended $\operatorname{val}$ function

In this section, we prove Theorem 1.1. The most important point of the proof below is the following lemma which is based on the proof of [BI19, Theorem 4.3] and is a generalization of [Mur20, Lemma 3.3].

Lemma 7.1 (Repetition frequency estimation).

Let $\{a_{1,n}\}_{n=1}^{\infty}$ , $\dots$ , $\{a_{k,n}\}_{n=1}^{\infty}$ be sequences in $\mathbb{Z}_{\geq 0}$ such that $a_{i,n}\to\infty$ and $2^{-n}a_{i,n}\to 0$ as $n\to\infty$ . Let $V$ be a non-empty even word and put $v:=[\overline{V}]$ . Let $V_{n}$ and $W_{n}$ be non-empty even words such that $V_{n}$ and $V_{n+1}$ , $W_{n}$ and $W_{n+1}$ coincide in the first $n$ terms respectively. Let $\{x_{n}\}_{n=1}^{\infty}$ and $\{x^{\prime}_{n}\}_{n=1}^{\infty}$ be sequences of real numbers whose continued fraction expansions are

(7.1)

x_{n}=[V^{a_{n}}V_{n}\cdots],\quad-x^{\prime}_{n}=[0,W_{n}\cdots]

respectively. Then

(7.2)

I_{n}:=\int_{z_{0}}^{\gamma_{V}^{a_{n}}z_{0}}j\eta_{x^{\prime}_{n},x_{n}}-a_{n}N(V)\widehat{\operatorname{val}}(v)

is a Cauchy sequence. In particular, we have

(7.3)

\int_{z_{0}}^{\gamma_{V}^{a_{n}}z_{0}}j\eta_{x^{\prime}_{n},x_{n}}=a_{n}N(V)\widehat{\operatorname{val}}(v)+O(1)\quad\text{ as }n\to\infty.

Proof.

Take positive integers $m<n$ with $a_{m}<a_{n}$ . We have

(7.4)

I_{n}-I_{m}=\int_{z_{0}}^{\gamma_{V}z_{0}}j\left(\sum_{0\leq i<a_{n}}\eta_{\gamma_{V}^{i-a_{n}}x^{\prime}_{n},\gamma_{V}^{i-a_{n}}x_{n}}-\sum_{0\leq i<a_{m}}\eta_{\gamma_{V}^{i-a_{m}}x^{\prime}_{m},\gamma_{V}^{i-a_{m}}x_{m}}+(a_{n}-a_{m})\eta_{v^{\prime},v}\right).

Let

(7.5)	$\displaystyle S_{1}(m,n;z)$	$\displaystyle:=\sum_{0\leq i<a_{m}}\left(\frac{1}{z-\gamma_{V}^{i-a_{n}}x_{n}}-\frac{1}{z-\gamma_{V}^{i-a_{m}}x_{m}}\right),$
(7.6)	$\displaystyle S^{\prime}_{1}(m,n;z)$	$\displaystyle:=\sum_{0\leq i<a_{m}}\left(\frac{1}{z-\gamma_{V}^{i-a_{n}}x^{\prime}_{n}}-\frac{1}{z-\gamma_{V}^{i-a_{m}}x^{\prime}_{m}}\right),$
(7.7)	$\displaystyle S_{2}(m,n;z)$	$\displaystyle:=\sum_{a_{m}\leq i<a_{n}}\left(\frac{1}{z-\gamma_{V}^{i-a_{n}}x_{n}}-\frac{1}{z-v}\right),$
(7.8)	$\displaystyle S^{\prime}_{2}(m,n;z)$	$\displaystyle:=\sum_{a_{m}\leq i<a_{n}}\left(\frac{1}{z-\gamma_{V}^{i-a_{n}}x^{\prime}_{n}}-\frac{1}{z-v^{\prime}}\right).$

Then we can write

(7.9)

I_{n}-I_{m}=\int_{z_{0}}^{\gamma_{V}z_{0}}j\left(-S_{1}(m,n;z)+S^{\prime}_{1}(m,n;z)+S_{2}(m,n;z)-S^{\prime}_{2}(m,n;z)\right)dz.

For each $0\leq i<a_{m}$ , since

	$\displaystyle\gamma_{V}^{i-a_{n}}x_{n}$	$\displaystyle=[V^{i}V_{n}\cdots],\quad$	$\displaystyle\gamma_{V}^{i-a_{m}}x_{m}$	$\displaystyle=[V^{i}V_{m}\cdots],$
	$\displaystyle\gamma_{V}^{i-a_{n}}x^{\prime}_{n}$	$\displaystyle=-[0,V^{\prime i}W_{n}\cdots],\quad$	$\displaystyle\gamma_{V}^{i-a_{m}}x^{\prime}_{m}$	$\displaystyle=-[0,V^{\prime i}W_{m}\cdots],$

we have

(7.10)

\left\lvert\gamma_{V}^{i-a_{n}}x_{n}-\gamma_{V}^{i-a_{m}}x_{m}\right\rvert\leq 2^{2-m},\quad\left\lvert\gamma_{V}^{i-a_{n}}x^{\prime}_{n}-\gamma_{V}^{i-a_{m}}x^{\prime}_{m}\right\rvert\leq 2^{2-m}.

For each $a_{m}\leq i<a_{n}$ , since

(7.11)

\gamma_{V}^{i-a_{n}}x_{n}=[V^{i}V_{n}\cdots],\quad\gamma_{V}^{i-a_{n}}x^{\prime}_{n}=-[0,V^{\prime i}W_{n}\cdots],

we have

(7.12)

\left\lvert\gamma_{V}^{i-a_{n}}x_{n}-v\right\rvert\leq 2^{2-a_{m}},\quad\left\lvert\gamma_{V}^{i-a_{n}}x^{\prime}_{n}-v^{\prime}\right\rvert\leq 2^{2-a_{m}}.

By Lemma 5.2, we obtain

(7.13)

\left\lvert I_{n}-I_{m}\right\rvert=\sum_{0\leq i<a_{m}}O(2^{-m})+\sum_{a_{m}\leq i<a_{n}}O(2^{-a_{m}})=O(a_{m}2^{-m})+O((a_{n}-a_{m})2^{-a_{m}}).

Since this converges to 0 as $m\to\infty$ by the assumption, $\{I_{n}\}$ is a Cauchy sequence. ∎

Proof of Theorem 1.1.

Keep the notation in Theorem 1.1 and let

(7.14)

L:=\lim_{n\to\infty}\frac{\left\lvert U_{1}\right\rvert+\cdots+\left\lvert U_{n}\right\rvert}{A_{n}}.

It suffices to show that

(7.15)

L\widehat{\operatorname{val}}(x)=a_{1}\widehat{\operatorname{val}}(W_{1})+\dots+a_{k}\widehat{\operatorname{val}}(W_{k}).

Take any real number $x^{\prime}<-1$ and for positive integers $n$ and $1\leq i\leq k$ , let

(7.16)	$\displaystyle x_{n}$	$\displaystyle:=\gamma_{U_{1}\cdots U_{n-1}}^{-1}x=[U_{n}U_{n+1}\cdots],$
(7.17)	$\displaystyle x^{\prime}_{n}$	$\displaystyle:=\gamma_{U_{1}\cdots U_{n-1}}^{-1}x^{\prime}=-[0,U^{\prime}_{n-1}\cdots U^{\prime}_{1},\delta_{0}x^{\prime}],$
(7.18)	$\displaystyle x_{i,n}$	$\displaystyle:=\gamma_{V_{1}W_{1}^{a_{1,n}}V_{2}W_{2}^{a_{2,n}}\cdots V_{i-1}W_{i-1}^{a_{i-1,n}}}^{-1}x_{n}=[V_{i}W_{i}^{a_{i,n}}\cdots V_{k}W_{k}^{a_{k,n}}U_{n+1}U_{n+2}\cdots],$
(7.19)	$\displaystyle x^{\prime}_{i,n}$	$\displaystyle:=\gamma_{V_{1}W_{1}^{a_{1,n}}V_{2}W_{2}^{a_{2,n}}\cdots V_{i-1}W_{i-1}^{a_{i-1,n}}}^{-1}x_{n}=-[0,{W^{\prime}}_{i-1}^{a_{i-1,n}}V^{\prime}_{i-1}\cdots{W^{\prime}}_{1}^{a_{1,n}}V^{\prime}_{1}U^{\prime}_{n-1}\cdots U^{\prime}_{1},\delta_{0}x^{\prime}].$

Since $x^{\prime}<-1$ , the far right-hand sides in the second and fourth rows are continued fraction expansions. Then we have

(7.20)	$\displaystyle L\widehat{\operatorname{val}}(x)$	$\displaystyle=\lim_{n\to\infty}\frac{1}{A_{n}}\sum_{1\leq m\leq n}\int_{\gamma_{U_{1}\cdots U_{m-1}}z_{0}}^{\gamma_{U_{1}\cdots U_{m}}z_{0}}j\eta_{x^{\prime},x}$
(7.21)		$\displaystyle=\lim_{n\to\infty}\frac{1}{A_{n}}\sum_{1\leq m\leq n}\int_{z_{0}}^{\gamma_{U_{m}}z_{0}}j\eta_{x^{\prime}_{m},x_{m}}$
(7.22)		$\displaystyle=\lim_{n\to\infty}\frac{1}{A_{n}}\sum_{1\leq m\leq n}\sum_{1\leq i\leq k}\left(\int_{z_{0}}^{\gamma_{V_{i}}z_{0}}j\eta_{x^{\prime}_{i,m},x_{i,m}}+\int_{z_{0}}^{W_{i}^{a_{i,m}}z_{0}}j\eta_{\gamma_{V_{i}}^{-1}x^{\prime}_{i,m},\gamma_{V_{i}}^{-1}x_{i,m}}\right).$

Since we have

(7.23)

\left\lvert x_{i,m}-\gamma_{V_{i}}w_{i}\right\rvert\leq 2^{2-a_{i,m}},\,\left\lvert x^{\prime}_{i,m}-\gamma_{w^{\prime}_{i-1}}\right\rvert\leq 2^{2-a_{i-1,m}}

by continued fraction expansion, it follows from Lemma 5.2 that

(7.24)

\int_{z_{0}}^{\gamma_{V_{i}}z_{0}}j\left(\eta_{x^{\prime}_{i,m},x_{i,m}}-\eta_{w^{\prime}_{i-1},\gamma_{V_{i}}w_{i}}\right)=O(2^{-a_{i,m}}+2^{-a_{i-1,m}}).

Since

(7.25)		$\displaystyle\gamma_{V_{i}}^{-1}x_{i,m}$	$\displaystyle=[W_{i}^{a_{i,m}}\cdots V_{k}W_{k}^{a_{k,m}}U_{n+1}U_{n+2}\cdots],$
(7.26)		$\displaystyle\gamma_{V_{i}}^{-1}x^{\prime}_{i,m}$	$\displaystyle=-[0,V^{\prime}_{i}{W^{\prime}}_{i-1}^{a_{i-1,m}}V^{\prime}_{i-1}\cdots{W^{\prime}}_{1}^{a_{1,m}}V^{\prime}_{1}U^{\prime}_{m-1}\cdots U^{\prime}_{1},\delta_{0}x^{\prime}],$

we have

(7.27)

\int_{z_{0}}^{W_{i}^{a_{i,m}}z_{0}}j\eta_{\gamma_{V_{i}}^{-1}x^{\prime}_{i,m},\gamma_{V_{i}}^{-1}x_{i,m}}=a_{i,m}N(W_{i})\widehat{\operatorname{val}}(w_{i})+O(1)

by Lemma 7.1. Thus, we obtain

(7.28)		$\displaystyle L\widehat{\operatorname{val}}(x)$	$\displaystyle=\lim_{n\to\infty}\frac{1}{A_{n}}\sum_{1\leq m\leq n}\sum_{1\leq i\leq k}a_{i,m}N(W_{i})\widehat{\operatorname{val}}(w_{i})$
(7.29)			$\displaystyle=\sum_{1\leq i\leq k}a_{i}N(W_{i})\widehat{\operatorname{val}}(w_{i}).$

∎

8. The value of $\operatorname{val}$ function at the Thue–Morse word

In this section, we calculate the value of $\operatorname{val}$ function at the Thue–Morse word and prove Theorem 1.2.

We fix two even words $V$ and $W$ . Let $h\colon\{V,W\}^{*}\to\{V,W\}^{*}$ be a monoid homomorphism such that $h(V):=VW,h(W):=WV$ and $V_{h}:=\lim_{n\to\infty}h^{n}(V)$ be the Thue–Morse word. For a word $U=U_{1}\cdots U_{n}\in\{V,W\}^{*}$ with $U_{1},\dots,U_{n}\in\{V,W\}$ , let $U^{\prime}:=U_{n}\cdots U_{1}$ .

To begin with, we give a simple representation of the Thue–Morse word.

Lemma 8.1.

Let $\tau\colon\{V,W\}^{*}\to\{V,W\}^{*}$ be a monoid homomorphism such that $\tau(V):=W$ and $\tau(W):=V$ . Then the following statements hold.

(i)

For any positive integer $n$ , we have $h^{n}(V)=h^{n-1}(V)\tau h^{n-1}(V)$ .
(ii)

$h^{2n}(V)^{\prime}=h^{2n}(V)$ .

Proof.

The first claim (i) follows from the facts that $h\circ\tau=\tau\circ h$ and $h^{2}(U)=h(U)\tau h(U)$ for a word $U\in\{V,W\}^{*}$ such that $h(U)=U\tau(U)$ . The last claim (ii) follows from the fact that $h^{2}(U)^{\prime}=h^{2}(U)$ for a word $U\in\{V,W\}^{*}$ such that $h(U)=U\tau(U)$ and $U^{\prime}=U$ . ∎

Proof of Theorem 1.2.

Let $V_{n}:=h^{n}(V)$ . Since $[V_{n}]^{\prime}=-[0,V_{n}]$ by Lemma 8.1 (ii), we have

(8.1)

\widehat{\operatorname{val}}([V_{h}])=\lim_{n\to\infty}\frac{1}{2^{2n}}\int_{z_{0}}^{\gamma_{V_{n}}z_{0}}j\eta_{-[0,V_{h}],V_{h}},\quad\widehat{\operatorname{val}}([V_{n}])=\int_{z_{0}}^{\gamma_{V_{n}}z_{0}}j\eta_{-[0,V_{n}],V_{n}}.

Here the first $2^{2n}$ terms of $V_{h}$ are $V_{n}$ by Lemma 8.1 (i). Thus, we obtain the claim. ∎

References

[Aig97] M. Aigner. Markov’s theorem and 100 years of the uniqueness conjecture. Springer, 1997.
[Bea83] A. F. Beardon. The geometry of discrete groups, volume 91 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1983.
[BI19] P. Bengoechea and Ö. Imamoglu. Cycle integrals of modular functions, Markov geodesics and a conjecture of Kaneko. Algebra Number Theory, 13(4):943––962, 2019.
[CK97] C. Choffrut and J. Karhumäki. Combinatorics of words. In Handbook of formal languages, Vol. 1, pages 329–438. Springer, Berlin, 1997.
[Dal11] F. Dal’Bo. Geodesic and horocyclic trajectories. Universitext. Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2011. Translated from the 2007 French original.
[DIT11] W. Duke, Ö. Imamoḡlu, and Á. Tóth. Cycle integrals of the $j$ -function and mock modular forms. Ann. of Math. (2), 173(2):947–981, 2011.
[Kan09] M. Kaneko. Observations on the ‘values’ of the elliptic modular function $j(\tau)$ at real quadratics. Kyushu Journal of Mathematics, 63(2):353–364, 2009.
[Mat20] T. Matsusaka. A hyperbolic analogue of the rademacher symbol. 2020. arXiv:2003.12354.
[Mur20] Y. Murakami. A continuity of cycle integrals of modular functions. Ramanujan J., 2020.
[Shi72] G. Shimura. Class fields over real quadratic fields and Hecke operators. Ann. of Math. (2), 95:130–190, 1972.
[Shi78] Takuro Shintani. On certain ray class invariants of real quadratic fields. J. Math. Soc. Japan, 30(1):139–167, 1978.
[Zag02] D. Zagier. Traces of singular moduli. In Motives, polylogarithms and Hodge theory, Part I (Irvine, CA, 1998), volume 3 of Int. Press Lect. Ser., pages 211–244. Int. Press, Somerville, MA, 2002.

Extended-cycle integrals of modular functions for badly approximable numbers

Abstract.

1. Introduction

Theorem 1.1.

Theorem 1.2.

Acknowledgement

2. A renewal definition of val\operatorname{val} for real quadratic numbers

2.1. Basic notation and properties for geodesics

Lemma 2.1.

Lemma 2.2 ([Mur20, Lemma 2.5]).

Lemma 2.3.

Proof.

2.2. Basic notation and properties for real quadratic numbers

2.3. Basic notation and properties for continued fractions and words

Lemma 2.4 ([Mur20, Lemma 2.3]).

2.4. Kaneko’s val

Lemma 2.5.

Lemma 2.6 ([Mur20, Propositon 2.7]).

Proposition 2.7.

Proof.

Proposition 2.8.

Proof.

Remark 2.9.

3. Fundamental properties of the extended val\operatorname{val} function

Theorem 3.1.

4. The extended val\operatorname{val} function as the limit value along a geodesic

Proposition 4.1 ([Dal11, Chapter VII, Theorem 3.4], [Dal11, Chapter VII, Lemma 3.5]).

Theorem 4.2 (Theorem 3.1 (i) and (ii) for val⁡(x)\operatorname{val}(x)).

Proof.

5. The extended val\operatorname{val} function as the limit value along a continued fraction

Lemma 5.1.

Proof.

Lemma 5.2.

Proof.

Theorem 5.3 (Theorem 3.1 (i) for val^​(x)\widehat{\operatorname{val}}(x) and 1^​(x)\widehat{1}(x)).

Proof.

Lemma 5.4.

Proof.

Proposition 5.5.

Theorem 5.6 (Theorem 3.1 (iii)).

Proof.

6. Elementary units for badly approximable numbers

Theorem 6.1 (Theorem 3.1 (iv) and (v)).

Remark 6.2.

Proof of Theorem 6.1.

Example 6.3.

7. An explicit computation of values of extended val\operatorname{val} function

Lemma 7.1 (Repetition frequency estimation).

Proof.

Proof of Theorem 1.1.

8. The value of val\operatorname{val} function at the Thue–Morse word

Lemma 8.1.

Proof.

Proof of Theorem 1.2.

References

2. A renewal definition of $\operatorname{val}$ for real quadratic numbers

3. Fundamental properties of the extended $\operatorname{val}$ function

4. The extended $\operatorname{val}$ function as the limit value along a geodesic

Theorem 4.2 (Theorem 3.1 (i) and (ii) for $\operatorname{val}(x)$ ).

5. The extended $\operatorname{val}$ function as the limit value along a continued fraction

Theorem 5.3 (Theorem 3.1 (i) for $\widehat{\operatorname{val}}(x)$ and $\widehat{1}(x)$ ).

7. An explicit computation of values of extended $\operatorname{val}$ function

8. The value of $\operatorname{val}$ function at the Thue–Morse word