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Remarks on uniform recurrence properties for beta-transformation

Yann Bugeaud

TL;DR

This note revisits uniform recurrence properties for the beta-transformation by analyzing the exponents $r_\beta$ and $\widehat{r}_\beta$ and the sets $R_\beta(\widehat{r},r)$. It corrects a claim from Zheng and Wu (2020) by showing, via Sturmian sequences, that there exist uncountably many $x$ with $\widehat{r}_\beta(x)=1$ and a wide range of $r_\beta(x)$, and it proves a universal bound $\widehat{r}_\beta(x)\le1$ for non-purely periodic expansions. The paper further proves that the set of $x$ with $\widehat{r}_\beta > r_\beta/(1+r_\beta)$ has Hausdorff dimension zero, and provides the exact dimension for metrical sets of interest, thereby preserving the overall metrical conclusions of prior work while clarifying the structure of these exceptional sets. The results employ a blend of beta-expansion combinatorics, Sturmian sequences, and covering arguments to connect Diophantine properties with symbolic dynamics in the beta-setting.

Abstract

We complement the recent paper of Zheng and Wu [Uniform recurrence properties for beta-transformation, Nonlinearity 33 (2020), 4590--4612], where the authors study, from the metrical point of view, the uniform recurrence properties of the orbit of a point under the $β$-transformation to the point itself.

Remarks on uniform recurrence properties for beta-transformation

TL;DR

This note revisits uniform recurrence properties for the beta-transformation by analyzing the exponents and and the sets . It corrects a claim from Zheng and Wu (2020) by showing, via Sturmian sequences, that there exist uncountably many with and a wide range of , and it proves a universal bound for non-purely periodic expansions. The paper further proves that the set of with has Hausdorff dimension zero, and provides the exact dimension for metrical sets of interest, thereby preserving the overall metrical conclusions of prior work while clarifying the structure of these exceptional sets. The results employ a blend of beta-expansion combinatorics, Sturmian sequences, and covering arguments to connect Diophantine properties with symbolic dynamics in the beta-setting.

Abstract

We complement the recent paper of Zheng and Wu [Uniform recurrence properties for beta-transformation, Nonlinearity 33 (2020), 4590--4612], where the authors study, from the metrical point of view, the uniform recurrence properties of the orbit of a point under the -transformation to the point itself.

Paper Structure

This paper contains 10 sections, 4 theorems, 41 equations.

Table of Contents

  1. Introduction
  2. Diophantine approximation and combinatorics on words
  3. Proof of Theorem \ref{['sturm']}
  4. Connection between the exponent $r_\beta$ and rational approximation, when $\beta$ is an integer
  5. Characteristic Sturmian sequences
  6. The Fibonacci sequence
  7. Completion of the proof of Theorem \ref{['sturm']}
  8. Proof of Theorem \ref{['bound']}
  9. Proof of Theorem \ref{['thdim']}
  10. Acknowledgements

Key Result

Theorem 1.2

Let $\beta > 1$ be a real number. There exists an uncountable set of real numbers ${\mathcal{R}}$ such that, for every $r$ in ${\mathcal{R}}$, there are uncountably many real numbers $x$ with ${\widehat{r}}_\beta (x) = 1$ and $r_\beta (x) = r$.

Theorems & Definitions (5)

  • Definition 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1