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Quantitative Matrix-Driven Diophantine approximation on $M_0$-sets

Bo Tan, Qing-Long Zhou

TL;DR

This work advances metric Diophantine approximation on $M_0$-sets by coupling a Rajchman-type Fourier-decay hypothesis with a matrix-dynamics framework driven by expanding integral matrices $A_n$. The authors prove a quantitative Schmidt-type counting theorem for approximations restricted to a fractal set and restricted to denominators from the matrix sequence, establishing equidistribution of $A_n\mathbf{x}$ for $\mu$-a.e. $\mathbf{x}$ and sharp shrinking-target counts. The approach extends Pollington–Velani–Zafeiropoulos–Zorin from the one-dimensional to the matrix setting via an $A$-divisibility mechanism, a higher-dimensional Davenport–Erdős–LeVeque criterion, and careful Fourier-analytic control of overlapping measures. Applications include normality results and shrinking target problems on fractals, providing a Khintchine-type dichotomy for hitting shrinking targets under matrix dynamics, with potential implications for normal numbers in fractal geometries and related metric Diophantine questions.

Abstract

Let $E\subset [0,1)^{d}$ be a set supporting a probability measure $μ$ with Fourier decay $|\widehatμ({\bf{t}})|\ll (\log |{\bf{t}}|)^{-s}$ for some constant $s>d+1.$ Consider a sequence of expanding integral matrices $\mathcal{A}=(A_n)_{n\in\N}$ such that the minimal singular values of $A_{n+1}A_{n}^{-1}$ are uniformly bounded below by $K>1$. We prove a quantitative Schmidt-type counting theorem under the following constraints: (1) the points of interest are restricted to $E$; (2) the denominators of the ``shifted'' rational approximations are drawn exclusively from $\mathcal{A}$. Our result extends the work of Pollington, Velani, Zafeiropoulos, and Zorin (2022) to the matrix setting, advancing the study of Diophantine approximation on fractals. Moreover, it strengthens the equidistribution property of the sequence $(A_n{\bf x})_{n\in\N}$ for $μ$-almost every ${\bf x}\in E.$ Applications include the normality of vectors and shrinking target problems on fractal sets.

Quantitative Matrix-Driven Diophantine approximation on $M_0$-sets

TL;DR

This work advances metric Diophantine approximation on -sets by coupling a Rajchman-type Fourier-decay hypothesis with a matrix-dynamics framework driven by expanding integral matrices . The authors prove a quantitative Schmidt-type counting theorem for approximations restricted to a fractal set and restricted to denominators from the matrix sequence, establishing equidistribution of for -a.e. and sharp shrinking-target counts. The approach extends Pollington–Velani–Zafeiropoulos–Zorin from the one-dimensional to the matrix setting via an -divisibility mechanism, a higher-dimensional Davenport–Erdős–LeVeque criterion, and careful Fourier-analytic control of overlapping measures. Applications include normality results and shrinking target problems on fractals, providing a Khintchine-type dichotomy for hitting shrinking targets under matrix dynamics, with potential implications for normal numbers in fractal geometries and related metric Diophantine questions.

Abstract

Let be a set supporting a probability measure with Fourier decay for some constant Consider a sequence of expanding integral matrices such that the minimal singular values of are uniformly bounded below by . We prove a quantitative Schmidt-type counting theorem under the following constraints: (1) the points of interest are restricted to ; (2) the denominators of the ``shifted'' rational approximations are drawn exclusively from . Our result extends the work of Pollington, Velani, Zafeiropoulos, and Zorin (2022) to the matrix setting, advancing the study of Diophantine approximation on fractals. Moreover, it strengthens the equidistribution property of the sequence for -almost every Applications include the normality of vectors and shrinking target problems on fractal sets.
Paper Structure (12 sections, 15 theorems, 118 equations)

This paper contains 12 sections, 15 theorems, 118 equations.

Table of Contents

  1. Introduction
  2. Equidistribution on $M_0$-sets in $\mathbb{R}$
  3. Equidistribution of vectors on $M_0$-sets
  4. Applications
  5. Normality of vectors on fractals
  6. Shrinking targets on fractals
  7. Proof of Theorem \ref{['TZ2']}
  8. Proof of Theorem \ref{['TZ']}
  9. $A$-divisibility
  10. Warm-up
  11. Expectation transference principle
  12. Overlapping Measures

Key Result

Theorem 1.2

Let $\mu$ be a Borel probability measure supported on a subset $E\subseteq[0,1].$ Let $\mathcal{A}=(q_n)_{n\in\mathbb{N}}$ be a sequence of reals satisfying for any $k\in \mathbb{Z}\setminus\{0\},$ then for $\mu$ almost all $x\in E$ the sequence $(q_nx)_{n\in \mathbb{N}}$ is equdistributed.

Theorems & Definitions (28)

  • Definition 1.1
  • Theorem 1.2: Davenport-Erdős-Leveque, DEL63
  • Theorem 1.3: Pollington-Velani-Zafeiropoulos-Zorin, PVZZ22
  • Remark 1.4
  • Remark 1.5
  • Theorem 1.6: Baker-Khalil-Sahlsten, BKS25
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Theorem 1.10
  • ...and 18 more