Sparse systems of functions and Quasi-analytic classes

Marco Aldi , Jeffrey Buffkin II , Cody Cline and Sean Cox

Abstract.

We provide a new characterization of quasi-analyticity of Denjoy-Carleman classes, related to Wetzel’s Problem. We also completely resolve which Denjoy-Carleman classes carry sparse systems: if the Continuum Hypothesis (CH) holds, all Denjoy-Carleman classes carry sparse systems; but if CH fails, a Denjoy-Carleman class carries a sparse system if and only if it is not quasi-analytic. As corollaries, we extend results of [MR3552748] and [CodyCoxLee] about non-existence of “anonymous predictors” for real functions.

2020 Mathematics Subject Classification:

03E50, 03E25, 26E05, 30D20

Partially funded by VCU SEED Grant and NSF grant DMS-2154141 of Cox.

1. Introduction

Motivated by some questions about “anonymous predictors” of Hardin-Taylor [MR3100500] and Bajpai-Velleman [MR3552748] and an old theorem of Erdős [MR168482] about Wetzel’s Problem, Cody-Cox-Lee [CodyCoxLee] introduced the following concept. In what follows, $\operatorname{\textup{Homeo}^{+}}(\mathbb{R})$ refers to the set of increasing bijections from $\mathbb{R}$ to $\mathbb{R}$ , and if $P\in\mathbb{R}^{2}$ , we use $x_{P}$ and $y_{P}$ to denote the first and second coordinate of $P$ , respectively.

Definition 1.

Suppose $\Gamma\subset\operatorname{\textup{Homeo}^{+}}(\mathbb{R})$ . A sparse $\boldsymbol{\Gamma}$ -system is a collection of functions

\left\{f_{P}\ :\ P\in\mathbb{R}^{2}\right\}

such that:

(1)

Each $f_{P}$ is a member of $\Gamma$ that passes through the point $P$ ; and
(2)

For every $u\in\mathbb{R}$ , both

$\left\{f_{P}(u)\ :\ P\in\mathbb{R}^{2}\text{ and }u\neq x_{P}\right\}$

and

$\left\{f^{-1}_{P}(u)\ :\ P\in\mathbb{R}^{2}\text{ and }u\neq y_{P}\right\}$

are countable.

If $\Gamma$ is not necessarily contained in $\operatorname{\textup{Homeo}^{+}}(\mathbb{R})$ , by “sparse $\Gamma$ system” we will mean a sparse $\Gamma\cap\operatorname{\textup{Homeo}^{+}}(\mathbb{R})$ system.

Clearly, if $\Gamma_{0}\subseteq\Gamma_{1}$ , then any sparse $\Gamma_{0}$ system is also a sparse $\Gamma_{1}$ system. It is known that:

•

There is a sparse $C^{\infty}$ system (Bajpai-Velleman [MR3552748]), and in fact a sparse $D^{\infty}$ system (Cox-Elpers [Cox_Elpers]).
•

The existence of a sparse analytic system is equivalent to the Continuum Hypothesis (Cody-Cox-Lee [CodyCoxLee]).

It is natural to wonder about other classes of functions. Following Mandelbrojt [MR0006354], given a compact interval $K$ of the reals and a sequence $\{M_{k}\}$ of positive real numbers, $\boldsymbol{C_{K}\{M_{k}\}}$ denotes the class of all $C^{\infty}$ functions $f:K\to\mathbb{R}$ such that there exist positive real numbers $\beta_{f}$ and $B_{f}$ such that

(*)

\forall k\geq 0\ \ \|f^{(k)}\|\leq\beta_{f}B_{f}^{k}M_{k}.

For functions with domain $\mathbb{R}$ , Rudin [MR0924157] and Hughes [MR0272965] used $\boldsymbol{C\{M_{k}\}}$ to denote the set of $C^{\infty}$ functions $f:\mathbb{R}\to\mathbb{R}$ such that there are positive $\beta_{f}$ and $B_{f}$ such that (* ‣ 1) holds (on all of $\mathbb{R}$ ). Observe that a function $f\in C\{M_{k}\}$ must be bounded by the constant $\beta_{f}M_{0}$ ; so in particular $C\{M_{k}\}$ is disjoint from $\text{Homeo}^{+}(\mathbb{R})$ and hence not appropriate for the study of sparse systems from Definition 1. So we will also consider the class $\boldsymbol{C_{\text{loc}}\{M_{k}\}}$ , which refers to the set of all $C^{\infty}$ functions $f:\mathbb{R}\to\mathbb{R}$ such that $f\restriction K\in C_{K}\{M_{k}\}$ for every compact interval $K\subset\mathbb{R}$ . Observe that for any sequence $\{M_{k}\}$ of positive reals,

(1)

\text{real polynomials}\subseteq C_{\text{loc}}\{M_{k}\}\subseteq C^{\infty}(\mathbb{R}).

For example, when $M_{k}=k!$ , $C_{\text{loc}}\{M_{k}\}$ is the class of real analytic functions, and $C\{M_{k}\}$ is the class of bounded real analytic functions that can be extended to a holomorphic function on a strip about the real axis in $\mathbb{C}$ (see Rudin [MR0924157], Theorem 19.9).

We shall refer to any class of the form $C_{\text{loc}}\{M_{k}\}$ as a locally Denjoy-Carleman class.¹¹1This naming convention is in line with the current literature. We could not locate who first introduced classes of the form $C_{K}\{M_{k}\}$ , though the Denjoy-Carleman theorem—which characterizes when such classes are quasi-analytic—answered a question that Hadamard had raised about such classes in 1912 (Mandelbrojt [MR0006354]). Our paper addresses the question:

Question 2.

Which locally Denjoy-Carleman classes carry sparse systems?

We completely answer this question, but the answer depends on whether or not the Continuum Hypothesis (CH) holds:

(1)

CH implies that every locally Denjoy-Carleman class carries a sparse system.
(2)

$\neg\text{CH}$ implies that a locally Denjoy-Carleman class carries a sparse system if and only if it is not quasi-analytic (defined below).

These facts follow from Theorems 4 and 5 below. We recall the definition of quasi-analyticity. A class $\Gamma$ of infinitely differentiable functions is called quasi-analytic if the following implication holds for every $f\in\Gamma$ : if, for some $x_{0}$ , $f^{(k)}(x_{0})=0$ holds for all $k\geq 0$ , then $f$ is identically zero. Under the additional assumption that $\{M_{k}\}$ is log-convex, the Denjoy-Carleman Theorem [MR0924157] states that $C\{M_{k}\}$ is quasi-analytic if and only if $\sum_{k=1}^{\infty}M_{k-1}/M_{k}=\infty$ . We provide a new characterization of quasi-analyticity:

Theorem 3.

$C\{M_{k}\}$ is a non-quasi-analytic class if and only if it contains an uncountable family $F$ such that $\left|\left\{f(x)\ :\ f\in F\right\}\right|\leq 2$ for all $x\in\mathbb{R}$ .

The following theorem improves the Bajpai-Velleman [MR3552748] theorem that there is a sparse $C^{\infty}$ system.

Theorem 4.

All non-quasi-analytic locally Denjoy-Carleman classes carry sparse systems.

Our other main theorem characterizes the Continuum Hypothesis in terms of which locally Denjoy-Carleman classes carry sparse systems:

Theorem 5.

The following are equivalent:

(1)

The Continuum Hypothesis (CH)
(2)

Every locally Denjoy-Carleman class carries a sparse system.
(3)

At least one quasi-analytic locally Denjoy-Carleman class carries a sparse system.
(4)

The particular quasi-analytic locally Denjoy-Carleman class $C_{\text{loc}}\{k!\}$ —i.e., the class of real analytic functions—carries a sparse system.

The equivalence of (1) with (4) was proved in [CodyCoxLee]. The main new content here is the proof that (1) implies (2) (and the proof that (3) implies (1), though this is a trivial modification of Erdős [MR168482]).

$\Gamma$ -sparse systems were introduced by Cody-Cox-Lee [CodyCoxLee] mainly because, by an argument ultimately due to Bajpai and Velleman [MR3552748], the existence of such a system implies there is no good $\Gamma$ -anonymous predictor for functions from $\mathbb{R}$ into $\mathbb{R}$ (see sections 4 and 5 of [CodyCoxLee]). So Theorems 4 and 5 yield the following corollaries, respectively:

Corollary 6.

If $\Gamma$ is a non-quasi-analytic locally Denjoy-Carleman class, then there is no good $\Gamma$ -anonymous predictor for functions from $\mathbb{R}$ to $\mathbb{R}$ .

Corollary 7.

Assume CH. For any locally Denjoy-Carleman class $\Gamma$ , there is no good $\Gamma$ -anonymous predictor for functions from $\mathbb{R}$ to $\mathbb{R}$ .

Before continuing to the proofs of the above theorems, we observe that there is a kind of “lower frontier” (in the subgroup lattice of $\text{Homeo}^{+}(\mathbb{R})$ ) on which classes can carry sparse systems:

Lemma 8.

There is no sparse polynomial system.

Proof.

Assume toward a contradiction that

\left\{f_{P}\ :\ P=(x_{P},y_{P})\in\mathbb{R}^{2}\right\}

is a sparse polynomial system. Consider the subcollection

\mathcal{F}:=\left\{f_{(0,y)}\ :\ y\in\mathbb{R}\right\}.

By sparseness of the original system,

(2)

\forall x\neq 0\ \ Y_{x}:=\big{\{}f_{(0,y)}(x)\ :\ y\in\mathbb{R}\big{\}}\text{ is countable.}

For each non-negative integer $n$ , let $\mathcal{F}_{n}$ denote the set of members of $\mathcal{F}$ of degree $n$ . Since $\mathcal{F}$ is uncountable, there exists some $n^{*}$ such that $\mathcal{F}_{n^{*}}$ is uncountable.

Since $\mathcal{F}_{n^{*}}\subseteq\mathcal{F}$ , if we consider any value $x_{1}\neq 0$ , then

\left\{f(x_{1})\ :\ f\in\mathcal{F}_{n^{*}}\right\}

is contained in $Y_{x_{1}}$ , which is countable. So $\mathcal{F}_{n^{*}}$ can be partitioned into countably many subsets indexed by their outputs at $x_{1}$ . Therefore, there exists some value $y_{1}$ such that

G_{1}:=\left\{f\in\mathcal{F}_{n^{*}}\ :\ f(x_{1})=y_{1}\right\}

is uncountable.

Now choose any $x_{2}\notin\{0,x_{1}\}$ and again using (2), find a $y_{2}$ such that

G_{2}:=\left\{f\in G_{1}\ :\ f(x_{2})=y_{2}\right\}

is uncountable. All members of $G_{2}$ have the same outputs at $x_{1}$ and $x_{2}$ . If we continue this process with any distinct nonzero numbers $x_{1},x_{2},\dots,x_{n^{*}+1}$ , then we obtain a set $G_{n^{*}+1}$ of uncountably many polynomials of degree $n^{*}$ , all passing through the $n^{*}+1$ points

\big{\{}(x_{i},y_{i})\ :\ 1\leq i\leq n^{*}+1\big{\}}.

This violates the uniqueness of the Lagrange interpolating polynomial, yielding a contradiction. ∎

We record the following basic fact, that to check membership of $f$ in $C_{K}\{M_{k}\}$ for a compact interval $K$ , it suffices to find $\beta_{f}$ and $B_{f}$ that work for all sufficiently large derivatives:

Lemma 9.

Suppose $K$ is a compact interval of $\mathbb{R}$ , and $\{M_{k}\}$ is a sequence of positive real numbers. Suppose $N$ is a fixed natural number, $f\in C^{\infty}(K)$ , and there exist $\beta_{f}$ and $B_{f}$ such that

\forall k\geq N\ \ \|f^{(k)}\|\leq\beta_{f}B_{f}^{k}M_{k}.

Then $f\in C_{K}\{M_{k}\}$ .

Proof.

Since $K$ is compact and $f\in C^{\infty}(K)$ ,

\beta_{2}:=\text{max}\left\{\frac{\|f^{(k)}\|}{B_{f}^{k}M_{k}}\ :\ 0\leq k<N\right\}

is finite. Let $\widetilde{\beta}:=\text{max}(\beta_{f},\beta_{2})$ . Then $\|f^{(k)}\|\leq\widetilde{\beta}B_{f}^{k}M_{k}$ for all $k\geq 0$ . ∎

We also will use:

Lemma 10.

Given any (not necessarily log-convex) sequence $\{M_{k}\}$ of positive reals: $C\{M_{k}\}$ is quasi-analytic if and only if $C_{\text{loc}}\{M_{k}\}$ is quasi-analytic.

Proof.

$C\{M_{k}\}\subseteq C_{\text{loc}}\{M_{k}\}$ , so quasi-analyticity of $C_{\text{loc}}\{M_{k}\}$ trivially implies quasi-analyticity of $C\{M_{k}\}$ .

Now suppose $C_{\text{loc}}\{M_{k}\}$ is not quasi-analytic, as witnessed by a nonzero $f\in C_{\text{loc}}\{M_{k}\}$ such that (WLOG) $f^{(k)}(0)=0$ for all $k\geq 0$ . Let $I$ be a sufficiently large closed interval including 0, so that $f\restriction I$ is a nonzero function. Then $f\restriction I$ witnesses that $C_{I}\{M_{k}\}$ is not quasi-analytic. By Mandelbrojt [MR0006354] (see also Cohen [MR0225957]) it follows that the log-convexification $\{M^{\prime}_{k}\}$ of $\{M_{k}\}$ has the property that $\sum_{k=1}^{\infty}M^{\prime}_{k-1}/M^{\prime}_{k}<\infty$ . Then by the version of the Denjoy-Carleman Theorem from Rudin [MR0924157], $C\{M_{k}^{\prime}\}$ is not quasi-analytic. The lemma then follows since $M_{k}^{\prime}\leq M_{k}$ for all $k$ . ∎

2. New characterization of quasi-analyticity

The following lemma is adapted to the quasi-analytic case from Erdős [MR168482].

Lemma 11.

Let $f_{1},f_{2},f_{3}$ be functions in a quasi-analytic class $C\{M_{k}\}$ . Then the subset $S$ of all real numbers $x$ such that $|\{f_{1}(x),f_{2}(x),f_{3}(x)\}|<3$ is discrete.

Proof.

Let $S_{12}$ be the set of all $x\in\mathbb{R}$ such that $f_{1}(x)=f_{2}(x)$ . Suppose $S_{12}$ contains a convergent sequence $\{x_{n}\}$ . Then, by repeated application of the Mean Value Theorem, we conclude that all derivatives of $f_{1}-f_{2}\in C\{M_{k}\}$ vanish at $\lim_{n}x_{n}$ . Hence $f_{1}=f_{2}$ . Repeating this argument twice we conclude that the similarly defined subsets $S_{13}$ and $S_{23}$ are also discrete. Hence $S=S_{12}\cup S_{13}\cup S_{23}$ is discrete. ∎

The following lemma can be thought of as a sharpening of a result attributed to Lyndon by Erdős [MR168482].

Lemma 12.

If $C\{M_{k}\}$ is not quasi-analytic, then it contains an uncountable subset $F$ such that $\left|\left\{f(x)\ :\ f\in F\right\}\right|\leq 2$ for all $x\in\mathbb{R}$ .

Proof.

Let $\mathcal{C}\subseteq[0,1]$ be the standard Cantor set. By a theorem of Hughes [MR0272965], there exists $g\in C\{M_{k}\}$ such that

(1)

$g^{(n)}(x)=0$ for all $n\geq 0$ and for all $x\in\mathcal{C}\cup(1,\infty)\cup(-\infty,0)$ ;
(2)

$g(x)\neq 0$ for all $x\in(0,1)\setminus\mathcal{C}$ .

For each $a,b\in\mathcal{C}$ such that $a<b$ , let $f_{ab}:\mathbb{R}\to\mathbb{R}$ be the function such that $f_{ab}(x)=g(x)$ if $x\in[a,b]$ and $f_{ab}(x)=0$ otherwise. Then the collection $F$ of all $f_{ab}$ as above satisfies the statement of the lemma. ∎

The proof of Theorem 3 follows easily by combining Lemma 11 and Lemma 12.

3. Sparse systems for non-quasi-analytic classes

Our proof of Theorem 4 follows the outline of the proof in section 4 of Bajpai-Velleman. First, we need a lemma that yields extremely flat bump functions in $C\{M_{k}\}$ , provided the sequence $\{M_{k}\}$ grows sufficiently fast.

Lemma 13.

Assume $I$ is an open interval of real numbers, $\varepsilon>0$ , and $\{M_{k}\}$ a sequence of positive real numbers such that

(3)

\sum_{k=1}^{\infty}M_{k-1}/M_{k}<\infty.

Then there is a $C^{\infty}$ function $b$ such that $b(x)>0$ for all $x\in I$ , $b(x)=0$ for all $x\in\mathbb{R}\setminus I$ , and

\forall k\geq 0\ \ \|b^{(k)}\|\leq\varepsilon^{k}M_{k}.

Proof.

Define an auxilliary sequence

L_{n}:=\begin{cases}\frac{8M_{0}}{\varepsilon}&n=1\\ M_{n}&n\neq 1\end{cases}

First we observe that, to prove the lemma, it suffices to find a $C^{\infty}$ function $b$ that is positive on $I$ , zero on $\mathbb{R}\setminus I$ , and with the property that for some positive constant $\beta$ ,

(4)

\forall n\ \ ||b^{(n)}||\leq\beta\varepsilon^{n}L_{n}.

Indeed, set $\alpha:=\max\left\{1,\frac{8}{\varepsilon}\right\}$ . Then $L_{n}\leq\alpha M_{n}$ for every $n$ . If we can manage to find a $C^{\infty}$ bump function $b$ with support $I$ satisfying (4), then the function $\frac{1}{\beta\alpha}b$ will satisfy the conclusion of the lemma.

Let $E:=\mathbb{R}\setminus I$ , which is a closed set of reals. The assumption (3) and definition of $\{L_{n}\}$ ensures that

\sum_{n=1}^{\infty}L_{n-1}/L_{n}<\infty.

Hughes [MR0272965] demonstrates (with our $L_{n}$ ’s playing the role of his $M_{n}$ ’s) that, given any positive constant $\rho$ , if we set

\lambda_{\rho}:=\frac{L_{0}}{L_{1}}+\rho\ \underbrace{\sum_{n=2}^{\infty}\frac{L_{n-1}}{L_{n}}\left(\sum^{\infty}_{k=n}\frac{L_{k-1}}{L_{k}}\right)^{-\frac{1}{2}}}_{=:D}

then there exists a function $f_{\rho}\in C^{\infty}(\mathbb{R})$ which is zero on $E$ , and is positive on $E^{\mathsf{c}}(=I)$ , such that $\forall n\in\mathbb{N},||f_{\rho}^{(n)}||_{\infty}\leq 2(4\lambda_{\rho})^{n}L_{n}$ . In particular, if we set

\rho:=\frac{\varepsilon}{8D},

then $\lambda_{\rho}=\frac{\varepsilon}{4}$ . Therefore, $||f_{\rho}^{(n)}||_{\infty}\leq 2\varepsilon^{n}L_{n}$ for every $n\geq 0$ , yielding the desired (4). ∎

Corollary 14.

Suppose $I=(\ell,r)$ , $\varepsilon>0$ , and $\{M_{k}\}$ are as in the assumptions of Lemma 13. Then there exists an infinitely differentiable function $s:[\ell,r]\to\mathbb{R}$ such that:

(1)

$s$ is strictly increasing
(2)

$s(\ell)=0$
(3)

For all $k\geq 1$ : $s^{(k)}(\ell)=0=s^{(k)}(r)$ (where these are right and left derivatives, respecively)
(4)

for all $k\geq 0$ : $\|s^{(k)}\|\leq\varepsilon^{k}M_{k}$ .

Proof.

Let $b$ be as in the conclusion of Lemma 13, and define $s_{0}$ on $[\ell,r]$ by

s_{0}(x):=\int_{0}^{x}b(t)dt.

Since (on $[\ell,r]$ ) $s^{(k)}=b^{(k-1)}$ for all $k\geq 1$ , the properties of $b$ ensure that $s_{0}$ has all of the required properties, except possibly requirement (4). Note that the properties of $b$ ensure that

(5)

\forall k\geq 1\ \|s_{0}^{(k)}\|=\|b^{(k-1)}\|\leq\varepsilon^{k-1}M_{k-1}.

Assumption (3) implies that the terms $M_{k-1}/M_{k}$ converge to zero, so there is a natural number $N$ such that

\forall k\geq N\ \ \frac{M_{k-1}}{M_{k}}<\varepsilon=\frac{\varepsilon^{k}}{\varepsilon^{k-1}}

and hence

\forall k\geq N\ \ \|s_{0}^{(k)}\|\leq\varepsilon^{k-1}M_{k-1}<\varepsilon^{k}M_{k}.

Set

A:=\text{max}\left\{\frac{\|s_{0}^{(k)}\|}{\varepsilon^{k}M_{k}}\ :\ 0\leq k<N\right\}

If $A\leq 1$ we set $s:=s_{0}$ , and if $A>1$ , set $s:=\frac{1}{A}s_{0}$ . Then $s$ has the required properties.

∎

We are now ready to prove Theorem 4. For the rest of this section, assume $C_{\text{loc}}\{M_{k}\}$ is not quasi-analytic. Then by Lemma 10, $C\{M_{k}\}$ is not quasi-analytic. So by Rudin [MR0924157] we may WLOG assume that $\{M_{k}\}$ is log-convex. Then, by the Denjoy-Carleman Theorem,

\sum_{k=1}^{\infty}\frac{M_{k-1}}{M_{k}}<\infty.

For each rational $\Delta>0$ and each positive integer $i$ , fix a function

s_{\Delta,i}:[0,\Delta]\to\mathbb{R}

satisfying the conclusion of Corollary 14, with $I=[0,\Delta]$ and $\varepsilon=1/i$ . Define

y(\Delta,i):=s_{\Delta,i}(\Delta).

So, $s_{\Delta,i}$ has the following properties:

(1)

It maps $[0,\Delta]$ onto $[0,y(\Delta,i)]$ in a strictly increasing fashion;
(2)

All derivatives of $s_{\Delta,i}$ at the endpoints of the interval $[0,\Delta]$ are zero;
(3)

For all $k\geq 0$ , $\|s_{\Delta,i}^{(k)}\|\leq(1/i)^{k}M_{k}$ .

For each pair $p,q$ of rational numbers, each pair $\Gamma$ , $\Delta$ of positive rationals, and each positive integer $i$ , define

t_{p,q,\Delta,\Gamma,i}:[p,p+\Delta]\to\mathbb{R}

t_{p,q,\Delta,\Gamma,i}(x):=q+\frac{\Gamma}{y(\Delta,i)}s_{\Delta,i}(x-p)

Notice that $t_{p,q,\Delta,\Gamma,i}$ maps the interval $[p,p+\Delta]$ onto $[q,q+\Gamma]$ , and its derivatives at the endpoints are all zero.

Let $\mathcal{T}$ denote the set of all such $t_{p,q,\Delta,\Gamma,i}$ (for $p,q,\Delta,\Gamma\in\mathbb{Q}$ and $i\in\mathbb{N}$ ) that have the property

(6)

\frac{\Gamma}{y(\Delta,i)}\leq 1.

Notice that $\mathcal{T}$ is a countable set of functions, and the inequality (6) ensures that

(7)

t_{p,q,\Delta,\Gamma,i}\in\mathcal{T}\ \implies\ \ \forall k\geq 1\ \|t^{(k)}_{p,q,\Delta,\Gamma,i}\|\leq\|s_{\Delta,i}^{(k)}\|\leq(1/i)^{k}M_{k}.

Fix a point $P=(x_{P},y_{P})$ in the real plane; we need to define the function $h_{P}$ that will be part of our sparse $C_{\text{loc}}\{M_{k}\}$ system.

Fix any increasing sequence $\{p_{i}\}$ of rational numbers converging to $x_{P}$ , and set $\Delta_{i}:=p_{i+1}-p_{i}$ . Recursively define an increasing sequence $\{q_{i}\}$ of rational numbers converging to $y_{P}$ such that

(8)

\forall i\ \ y(\Delta_{i},i)>y_{P}-q_{i}>0.

Let $\Gamma_{i}:=q_{i+1}-q_{i}$ . It follows that

(9)

\forall i\ \ \Gamma_{i}=q_{i+1}-q_{i}<y_{P}-q_{i}<y(\Delta_{i},i)

and hence $\displaystyle\frac{\Gamma_{i}}{y(\Delta_{i},i)}\leq 1$ for all $i$ . So

(10)

\forall i\geq 0\left(t_{p_{i},q_{i},\Delta_{i},\Gamma_{i},i}\in\mathcal{T}\text{ and }\forall k\geq 1\left(\ \|t^{(k)}_{p_{i},q_{i},\Delta_{i},\Gamma_{i},i}\|\leq(1/i)^{k}M_{k}\right)\right).

Notice that for each $i$ , $t_{p_{i},q_{i},\Delta_{i},\Gamma_{i},i}$ meets $t_{p_{i+1},q_{i+1},\Delta_{i+1},\Gamma_{i+1},{i+1}}$ at the point $(p_{i+1},q_{i+1})$ , and both have all vanishing derivatives at $p_{i+1}$ . So their union is a strictly increasing $C^{\infty}$ function mapping $[p_{i},p_{i+2}]$ onto $[q_{i},q_{i+2}]$ . Furthermore, since $(p_{i},q_{i})$ converges to $P=(x_{P},y_{P})$ , the function $h:[p_{0},x_{P}]\to[q_{0},y_{P}]$ defined (as a set of ordered pairs) by

h:=\big{\{}(x_{P},y_{P})\big{\}}\cup\bigcup_{i\in\mathbb{N}}t_{p_{i},q_{i},\Delta_{i},\Gamma_{i},i}

is a strictly increasing function that is continuous on $[p_{0},x_{P}]$ and infinitely differentiable on $[p_{0},x_{P})$ . To see that it is infinitely differentiable at $x_{P}$ too, fix a $k\geq 1$ . By (10), we have

\lim_{i\to\infty}\|\underbrace{t^{(k)}_{p_{i},q_{i},\Delta_{i},\Gamma_{i},i}}_{h^{(k)}\restriction[p_{i},p_{i+1}]}\|=0,

so $\lim_{z\nearrow x_{P}}h^{(k)}(z)=0$ . By Lemma 6 of Bajpai-Velleman [MR3552748], the $k$ -th left derivative of $h$ at $x_{P}$ exists and is zero.

Now (10) also ensures that

(11)

\forall k\geq 1\ \|h^{(k)}\|\leq M_{k},

and so by Lemma 9, $h\in C_{[p_{0},x_{P}]}\{M_{k}\}$ .

This completes the heart of the construction, but the function we’ve constructed has domain $[p_{0},x_{P}]$ , not $\mathbb{R}$ . But still using members of $\mathcal{T}$ , we can easily extend $h$ to a strictly increasing $h_{L}:(-\infty,x_{P}]\to(-\infty,y_{P}]$ such that for all $k\geq 1$ , $\|h_{L}^{(k)}\|\leq M_{k}$ , as follows. For $n\in\mathbb{N}$ set $u_{n}:=p_{0}-n$ , $v_{n}:=q_{0}-n\cdot y(1,1)$ , and use the function $t_{u_{n+1},v_{n+1},1,y(1,1),1}$ to map $[u_{n+1},u_{n}]$ onto $[v_{n+1},v_{n}]$ . Combining with the earlier function on $[p_{0},x_{P}]$ , this yields a function mapping $(-\infty,x_{P}]$ onto $(-\infty,y_{P}]$ with the desired properties. And what we’ve done so far can obviously be done “from the right” of the point $P=(x_{P},y_{P})$ as well, yielding a function with similar properties on $[x_{P},\infty)$ . Thus we obtain an increasing bijection $h_{P}:\mathbb{R}\to\mathbb{R}$ passing through the point $P=(x_{P},y_{P})$ , such that

\forall k\geq 1\ \|h_{P}^{(k)}\|\leq M_{k}.

In particular, $h_{P}\in C_{\text{loc}}\{M_{k}\}$ .

Finally, we verify the sparseness of the system $\big{\langle}h_{P}\ :\ P=(x_{P},y_{P})\in\mathbb{R}^{2}\big{\rangle}$ . Fix any $u\in\mathbb{R}$ , and consider the collection of all points $P$ such that $u\neq x_{P}$ . By construction, for any such $P$ , $h_{P}(u)=t(u)$ for some $t$ in the countable set $\mathcal{T}$ ; so

\left\{h_{P}(u)\ :\ P\in\mathbb{R}^{2}\text{ and }u\neq x_{P}\right\}\subseteq\left\{\vphantom{t^{-1}(u)}t(u)\ :\ t\in\mathcal{T}\right\}

and the right side is countable, since $\mathcal{T}$ is countable. Similarly,

\left\{h^{-1}_{P}(u)\ :\ P\in\mathbb{R}^{2}\text{ and }u\neq y_{P}\right\}\subseteq\left\{t^{-1}(u)\ :\ t\in\mathcal{T}\right\}

is countable.

4. CH and sparse systems for all locally Denjoy-Carleman classes

We prove Theorem 5. Equivalence of (1) with (4) was proved in [CodyCoxLee]. (2) trivially implies (3), and (4) trivially implies (3). It remains to prove that (3) implies (1) and that (1) implies (2).

To see that (3) implies (1), suppose $\left\{f_{P}\ :\ P\in\mathbb{R}^{2}\right\}$ is a sparse $C_{\text{loc}}\{M_{k}\}$ system, where $C_{\text{loc}}\{M_{k}\}$ is quasi-analytic. Then

\left\{f_{(0,y)}\ :\ y\in\mathbb{R}\right\}

is an uncountable family of (restrictions of) functions from $C_{\text{loc}}\{M_{k}\}$ , all with domain $(-\infty,0)$ , which by sparseness of the original system has what Erdős [MR168482] calls “property $P_{0}$ ”. The quasi-analyticity ensures that if $f$ and $g$ are distinct members of $C_{\text{loc}}\{M_{k}\}$ , then $\left\{x\in\mathbb{R}\ :\ f(x)=g(x)\right\}$ is discrete, hence countable. The failure of CH then follows by exactly the same arguments as Erdős [MR168482].

Now we prove (1) implies (2). We first prove the following ZFC theorem, which strengthens Theorem 5 of [CodyCoxLee]:

Theorem 15.

Suppose $\mathcal{E}$ is a partition of $\mathbb{R}$ into dense subsets of $\mathbb{R}$ ; for each $z\in\mathbb{R}$ , let $E_{z}$ denote the unique member of $\mathcal{E}$ containing $z$ .

Then for any $P=(x_{P},y_{P})\in\mathbb{R}^{2}$ , any countable set $W$ of reals, and any sequence $\{N_{k}\}$ of positive reals, there is an entire $f:\mathbb{C}\to\mathbb{C}$ such that:

(1)

$f\restriction\mathbb{R}$ is an increasing bijection from the reals to the reals with strictly positive derivative;
(2)

$f(x_{P})=y_{P}$ ;
(3)
for each $w\in W$ :
- •
  
  if $w\neq x_{P}$ then $f(w)\in E_{w}$ ;
- •
  
  if $w\neq y_{P}$ then $f^{-1}(w)\in E_{w}$ ; and
(4)

$f\restriction\mathbb{R}\in C_{\text{loc}}\{N_{k}\}$ .

Proof.

We slightly modify the proof of Theorem 5 from [CodyCoxLee]. In that proof, the desired $f$ was the limit of a recursively-constructed sequence $\{f_{n}\}$ of polynomials, where $f_{0}(z)=\frac{3}{2}(z-x_{P})+y_{P}$ , and for each $n\geq 1$ , $f_{n}$ was of the form

f_{n}=f_{n-1}+\alpha_{n}M_{n}h_{n}

for some (recursively-defined) polynomial $h_{n}$ , some positive real $\alpha_{n}$ , and some $M_{n}\in[0,1]$ , defined in that order. The proof maintains a list of 7 inductive clauses, labelled (I)_n through (VII)_n, which ensure that the sequence $\{f_{n}\}$ was uniformly Cauchy on all compact $D\subset\mathbb{C}$ , and that the limit²²2Which is uniform on every compact $D\subset\mathbb{C}$ . of the $f_{n}$ ’s satisfies all the properties listed in Theorem 15, except possibly our new clause 4.

We will modify the recursion from [CodyCoxLee] ever so slightly; essentially at stage $n$ of the recursion, we define $h_{n}$ exactly as in [CodyCoxLee], but then shrink $\alpha_{n}$ even further; this has no effect on the maintenance of the inductive clauses (I)_n through (VII)_n. We also recursively construct another auxilliary sequence $\{B_{n}\}$ of (typically large) positive real numbers, and add another inductive clause to the seven already listed in [CodyCoxLee]:

\text{(VIII)}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}n}}:\ \ \ \forall i\ \forall j\ \ \left(i\leq j\leq{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}n}\ \implies\ \forall k\geq 1\ \ \left\lVert f_{j}^{(k)}\restriction D_{i}\right\rVert<B_{i}^{k}N_{k}\right)

Here, $D_{i}$ denotes the closed disc in $\mathbb{C}$ of radius $i$ centered at the origin. Note that since each $f_{j}$ is a polynomial, the inequality $\left\lVert f_{j}^{(k)}\restriction D_{i}\right\rVert<B_{i}^{k}N_{k}$ trivially holds for $k\geq\text{degree}(f_{j})$ , so the content of (VIII)_n is really about a finite collection of values of $k$ .

Suppose $\alpha_{n-1}$ , $M_{n-1}$ , $h_{n-1}$ , and $B_{n-1}$ have been defined, and that (I)_n-1 through (VII)_n-1 hold, along with our new inductive clause

\text{(VIII)}_{{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}n-1}}:\ \ \ \forall i\ \forall j\ \ \left(i\leq j\leq{\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}n-1}\ \implies\ \forall k\geq 1\ \ \left\lVert f_{j}^{(k)}\restriction D_{i}\right\rVert<B_{i}^{k}N_{k}\right).

The inductive step of the proof is almost verbatim from [CodyCoxLee], with the following alterations. First, the polynomial $h_{n}$ is defined exactly as in [CodyCoxLee] (top of page 5). They then argue that for all sufficiently small choices of $\alpha_{n}>0$ , there is an $M_{n}\in[0,1]$ such that the polynomial

f_{n}:=f_{n-1}+\alpha_{n}M_{n}h_{n}

satisfies their inductive clauses (I)_n through (VII)_n. Using assumption (VIII)_n-1, we simply shrink $\alpha_{n}$ further, if necessary, to ensure that the following finitely many additional constraints are met:

(12)

\forall i\leq n-1\ \ \forall k\leq\text{degree}\big{(}f_{n-1}+\alpha_{n}h_{n}\big{)}:\ \big{\lVert}f_{n-1}\restriction D_{i}\big{\rVert}+\big{\lVert}\alpha_{n}h_{n}\restriction D_{i}\big{\rVert}<B_{i}^{k}N_{k}.

Then define $M_{n}\in[0,1]$ (based on choice of $\alpha_{n}$ ) exactly as in [CodyCoxLee]. Then by (12) and the fact that $M_{n}\in[0,1]$ , it follows that

\displaystyle\forall i\leq n-1\ \ \forall k\leq\text{degree}\left(\overbrace{f_{n-1}+\alpha_{n}M_{n}h_{n}}^{f_{n}:=}\right):\ \begin{split}\big{\lVert}f_{n}\restriction D_{i}\big{\rVert}&\leq\big{\lVert}f_{n-1}\restriction D_{i}\big{\rVert}+\big{\lVert}\alpha_{n}M_{n}h_{n}\restriction D_{i}\big{\rVert}\\ &<B_{i}^{k}N_{k}.\end{split}

Hence, $f_{n}$ satisfies the relevant requirements for all $i<n$ . To satisfy the requirement at $n$ , simply define $B_{n}$ to be any strict upper bound of the set

(13)

\Big{\{}\big{\lVert}f_{n}^{(k)}\restriction D_{n}\big{\rVert}\ :\ k\leq\text{degree}(f_{n})\Big{\}}.

Thus, we have achieved (VIII)_n.

Since the sequence of $f_{n}$ ’s satisfy clauses (I)_n through (VII)_n of [CodyCoxLee], by that theorem their limit $f$ satisfies all clauses of Theorem 15 listed before clause 4. And, the fact that our additional clause (VIII)_n holds for all $n$ ensures that

(14)

\forall i\in\mathbb{N}\ \forall k\geq 1\ \big{\lVert}f^{(k)}\restriction D_{i}\big{\rVert}\leq B_{i}^{k}N_{k}

This implies that $f\restriction D_{i}\in C_{D_{i}}\{N_{k}\}$ for all $i$ , and hence $f\in C_{\text{loc}}\{N_{k}\}$ .

∎

Now let $\{N_{k}\}$ be any sequence of positive reals. The existence of a sparse $C_{\text{loc}}\{N_{k}\}$ system under CH follows, mutatis mutandis, as in the proof of Theorem 2 of [CodyCoxLee] (with “analytic” replaced by “ $C_{\text{loc}}\{N_{k}\}$ ”, and Theorem 5 of [CodyCoxLee] replaced by our Theorem 15).

5. Concluding remarks and open questions

We summarize the current knowledge of sparse systems, and present some open questions. We know:

•

Classes of real polynomials never carry sparse systems (Lemma 8).
•
Non-quasi-analytic locally Denjoy-Carleman classes always carry sparse systems (Theorem 4).
- –
  
  Assuming CH, all locally Denjoy-Carleman classes carry sparse systems (Theorem 5);
- –
  
  Assuming $\neg\text{CH}$ , the locally Denjoy-Carleman classes that carry sparse systems are exactly the non-quasi-analytic ones (follows from Theorems 5 and 4).

This completely answers Question 2, and establishes some clear dividing lines between classes that carry sparse systems and those that don’t:

•

Under CH, there is a divide between classes of polynomials and locally Denjoy-Carleman classes; all of the latter carry sparse systems, while none of the former do.
•

If CH fails, there is a divide between quasi-analytic locally Denjoy-Carleman classes and the non-quasi-analytic ones; all of the latter carry sparse systems, while none of the former do.

What about other intermediate classes? Regarding the CH setting, there is no obvious natural candidate for classes between polynomials and locally Denjoy-Carleman. If we were to only require the sequence $\{M_{k}\}$ to be non-negative in the definition of the locally Denjoy-Carleman class $C_{\text{loc}}\{M_{k}\}$ , then if $M_{k}=0$ for at least one $k$ , $C_{\text{loc}}\{M_{k}\}$ would be contained in the polynomials and hence, by Lemma 8, could not carry a sparse system.

References

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