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A dimensional mass transference principle from balls to open sets and applications to dynamical Diophantine approximation

Yubin He

Abstract

The mass transference principle of Beresnevich and Velani is a powerful mechanism for determining the Hausdorff dimension/measure of $\limsup$ sets that arise naturally in Diophantine approximation. However, in the setting of dynamical Diophantine approximation, this principle often fails to apply effectively, as the radii of the balls defining the dynamical $\limsup$ sets generally depend on the orbit of the point $x$ itself. In this paper, we develop a dimensional mass transference principle that enables us to recover and extend classical results on shrinking target problems, particularly for the $β$-transformation and the Gauss map. Moreover, our result shows that the corresponding $\limsup$ sets have large intersection properties. A potentially interesting feature of our method is that, in many cases, shrinking target problems are closely related to finding an appropriate Gibbs measure, which may reveal new aspects of the link between thermodynamic formalism and dynamical Diophantine approximation.

A dimensional mass transference principle from balls to open sets and applications to dynamical Diophantine approximation

Abstract

The mass transference principle of Beresnevich and Velani is a powerful mechanism for determining the Hausdorff dimension/measure of sets that arise naturally in Diophantine approximation. However, in the setting of dynamical Diophantine approximation, this principle often fails to apply effectively, as the radii of the balls defining the dynamical sets generally depend on the orbit of the point itself. In this paper, we develop a dimensional mass transference principle that enables us to recover and extend classical results on shrinking target problems, particularly for the -transformation and the Gauss map. Moreover, our result shows that the corresponding sets have large intersection properties. A potentially interesting feature of our method is that, in many cases, shrinking target problems are closely related to finding an appropriate Gibbs measure, which may reveal new aspects of the link between thermodynamic formalism and dynamical Diophantine approximation.

Paper Structure

This paper contains 13 sections, 28 theorems, 214 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Proof of Theorem \ref{['t:MTP']}
  4. Construction of a subset of $\bigcup_{k= \ell}^\infty E_k\cap B$
  5. Hausdorff content bound of $\bigcup_{k= \ell}^\infty E_k\cap B$
  6. Application to $\beta$-transformations
  7. Definition and some basic properties
  8. Pressure function
  9. Application to shrinking target problems
  10. Applications to Gauss map
  11. Definition and some basic properties
  12. Pressure function
  13. Applications

Key Result

Theorem 1.1

Let $X$ be a compact metric space equipped with a $\delta$-Ahlfors regular measure $\mu$. Let $\{B(x_n,r_n)\}$ be a sequence of balls in $X$ with $r_n\to 0$ as $n\to\infty$. Suppose that Then, for any $\tau>1$,

Figures (4)

  • Figure 1: When $f$ is constant, $E_2$ consists of intervals of equal length.
  • Figure 2: When $f$ is not constant, $E_2$ consists of intervals of varying lengths.
  • Figure 3: When $f$ is constant, the enlarged intervals contained in $E_{2,s}$ are well-separated and therefore efficiently cover $[0,1]$.
  • Figure 4: When $f$ is not constant, the enlarged intervals contained in $E_{2,s}$ may overlap and therefore fail to cover $[0,1]$ efficiently.

Theorems & Definitions (56)

  • Theorem 1.1: Mass transference principle BeresnevichVelaniMTPann
  • Definition 1.2: Quasi-self-conformality
  • Definition 1.3: FalconerLIPintroduce
  • Theorem 1.4
  • Remark 1
  • Remark 2
  • Remark 3
  • Theorem 1.5
  • Remark 4
  • Remark 5
  • ...and 46 more