Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Khintchine dichotomy and Schmidt estimates for self-similar measures on $\mathbb{R}^d$

Timothée Bénard, Weikun He, Han Zhang

Abstract

We extend the classical theorems of Khintchine and Schmidt in metric Diophantine approximation to the context of self-similar measures on $\mathbb{R}^d$. For this, we establish effective equidistribution of associated random walks on $\text{SL}_{d+1}(\mathbb{R})/\text{SL}_{d+1}(\mathbb{Z})$. This generalizes our previous work which requires $d=1$ and restricts Schmidt-type counting estimates to approximation functions which decay fast enough. Novel techniques include a bootstrap scheme for the associated random walks despite algebraic obstructions, and a refined treatment of Dani's correspondence. Along the way, we also establish non-concentration properties of self-similar measures near algebraic subvarieties of $\mathbb{R}^d$.

Khintchine dichotomy and Schmidt estimates for self-similar measures on $\mathbb{R}^d$

Abstract

We extend the classical theorems of Khintchine and Schmidt in metric Diophantine approximation to the context of self-similar measures on . For this, we establish effective equidistribution of associated random walks on . This generalizes our previous work which requires and restricts Schmidt-type counting estimates to approximation functions which decay fast enough. Novel techniques include a bootstrap scheme for the associated random walks despite algebraic obstructions, and a refined treatment of Dani's correspondence. Along the way, we also establish non-concentration properties of self-similar measures near algebraic subvarieties of .

Paper Structure

This paper contains 19 sections, 56 theorems, 314 equations.

Table of Contents

  1. Introduction
  2. Notations
  3. Moment and finite approximation of self-similar measures
  4. Regularity of self-similar measures
  5. Hölder regularity near affine subspaces
  6. Hölder regularity near subvarieties
  7. Finite time consequences
  8. Effective recurrence of the $\mu$-walk
  9. Positive dimension
  10. Dimensional Bootstrap
  11. Non-concentration inequalities
  12. Linear multislicing
  13. Linearizing charts
  14. Dimension increment and bootstrap
  15. From high dimension to equidistribution
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $\lambda$ be a probability measure on $\mathop{\mathrm{Sim}}\nolimits(\mathbb{R}^d)$, and assume $\lambda$ is irreducible with finite exponential moment. Let $\sigma$ be a $\lambda$-stationary probability measure on $\mathbb{R}^d$. Let $\psi : \mathbb{N}\rightarrow \mathbb{R}_{\geq0}$ be a non-i Moreover, in the divergent case $\sum_{ q\in \mathbb{N}}\psi(q)^d=\infty$, we have the following as

Theorems & Definitions (115)

  • Example
  • Theorem 1.1: Khintchine and Schmidt for self-similar measures
  • Theorem 1.2: Effective equidistribution of expanding fractals
  • Theorem 1.3: Effective equidistribution of the $\mu$-random walk
  • Lemma 3.1: Finite moment
  • proof
  • Lemma 3.2: Finite approximation
  • proof
  • Lemma 3.3: Moment estimate on right trajectories
  • proof
  • ...and 105 more