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Slow polynomial mixing, dynamical Borel-Cantelli lemma and Hausdorff dimension of dynamical diophantine sets

Edouard Daviaud

TL;DR

The paper investigates the interplay between slow polynomial mixing and dynamical Borel-Cantelli phenomena, along with the Hausdorff dimension of dynamical Diophantine sets, within a skew-product system on $ abla^3$ endowed with Lebesgue measure. It constructs a specific $oldsymbol{eta}$ so that the skew-product $S_{oldsymbol{eta}}$ is $(9,n o C/n^{s})$-mixing for some $0\le s o 1$, and proves that the dynamical Borel-Cantelli dichotomy can fail at every point under polynomial mixing, while simultaneously establishing a sharp upper bound on the dimension of shrinking-target limsup sets, namely $ ext{dim}_H<3$ for the target rate $1/n^{1/3}$. The results leverage discrepancy estimates for toral rotations, dyadic-cube decompositions, and Diophantine-approximation techniques via the Diophantine linear type exponent $oldsymbol{ ext{γ}_ ext{ell}}$. The work thus identifies precise thresholds linking mixing rates to both Borel-Cantelli-type behavior and dimension theory, demonstrating optimality of the constructed example and contributing to the understanding of dynamical Diophantine approximation in higher dimensions.

Abstract

In this article, we establish optimality results regarding the dynamical Borel-Cantelli lemma and the the Hausdorff dimension of certain dynamical diophantine sets.

Slow polynomial mixing, dynamical Borel-Cantelli lemma and Hausdorff dimension of dynamical diophantine sets

TL;DR

The paper investigates the interplay between slow polynomial mixing and dynamical Borel-Cantelli phenomena, along with the Hausdorff dimension of dynamical Diophantine sets, within a skew-product system on endowed with Lebesgue measure. It constructs a specific so that the skew-product is -mixing for some , and proves that the dynamical Borel-Cantelli dichotomy can fail at every point under polynomial mixing, while simultaneously establishing a sharp upper bound on the dimension of shrinking-target limsup sets, namely for the target rate . The results leverage discrepancy estimates for toral rotations, dyadic-cube decompositions, and Diophantine-approximation techniques via the Diophantine linear type exponent . The work thus identifies precise thresholds linking mixing rates to both Borel-Cantelli-type behavior and dimension theory, demonstrating optimality of the constructed example and contributing to the understanding of dynamical Diophantine approximation in higher dimensions.

Abstract

In this article, we establish optimality results regarding the dynamical Borel-Cantelli lemma and the the Hausdorff dimension of certain dynamical diophantine sets.

Paper Structure

This paper contains 7 sections, 8 theorems, 77 equations.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. Recall on geometric measure theory
  4. Continued fraction, Diophantine linear type
  5. Statement of the main result
  6. Proof of the first item of Theorem \ref{['Mainthm']}
  7. Proof of the second item of Theorem \ref{['Mainthm']}

Key Result

Proposition 2.1

There exists $\alpha_1,\alpha_2 \in\mathbb{R}$ such that writing $(\frac{p_n}{q_n})_{n\in\mathbb{N}}$ the partial quotients of $\alpha_1$ and $(\frac{p'_n}{q'_n})_{n\in\mathbb{N}}$ the one of $\alpha_2,$ one has

Theorems & Definitions (12)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.1: Saussogalato
  • Theorem 2.2: Saussogalato
  • Theorem 2.3: Edergo
  • Theorem 2.4
  • Corollary 2.5
  • Theorem 3.1: Saussogalato, Proposition 17
  • Lemma 3.2
  • ...and 2 more