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A zero-one law for improvements to Dirichlet's theorem in arbitrary dimension

Andreas Strömbergsson, Shucheng Yu

Abstract

Let $ψ$ be a continuous decreasing function defined on all large positive real numbers. We say that a real $m\times n$ matrix $A$ is $ψ$-Dirichlet if for every sufficiently large real number $t$ one can find $\mathbf{p} \in \mathbb{Z}^m$, $\mathbf{q} \in \mathbb{Z}^n\setminus\{\mathbf{0}\}$ satisfying $\|A\mathbf{q}-\mathbf{p}\|^m< ψ(t)$ and $\|\mathbf{q}\|^n<t$. By removing a technical condition from a partial zero-one law proved by Kleinbock-Strömbergsson-Yu, we prove a zero-one law for the Lebesgue measure of the set of $ψ$-Dirichlet matrices provided that $ψ(t)<1/t$ and $tψ(t)$ is increasing. In fact, we prove the zero-one law in a more general situation with the monotonicity assumption on $tψ(t)$ replaced by a weaker condition. Our proof follows the dynamical approach of Kleinbock-Strömbergsson-Yu in reducing the question to a shrinking target problem in the space of lattices. The key new ingredient is a family of carefully chosen subsets of the shrinking targets studied by Kleinbock-Strömbergsson-Yu, together with a short-range mixing estimate for the associated hitting events. Our method also works for the analogous weighted problem where the relevant supremum norms are replaced by certain weighted quasi-norms.

A zero-one law for improvements to Dirichlet's theorem in arbitrary dimension

Abstract

Let be a continuous decreasing function defined on all large positive real numbers. We say that a real matrix is -Dirichlet if for every sufficiently large real number one can find , satisfying and . By removing a technical condition from a partial zero-one law proved by Kleinbock-Strömbergsson-Yu, we prove a zero-one law for the Lebesgue measure of the set of -Dirichlet matrices provided that and is increasing. In fact, we prove the zero-one law in a more general situation with the monotonicity assumption on replaced by a weaker condition. Our proof follows the dynamical approach of Kleinbock-Strömbergsson-Yu in reducing the question to a shrinking target problem in the space of lattices. The key new ingredient is a family of carefully chosen subsets of the shrinking targets studied by Kleinbock-Strömbergsson-Yu, together with a short-range mixing estimate for the associated hitting events. Our method also works for the analogous weighted problem where the relevant supremum norms are replaced by certain weighted quasi-norms.
Paper Structure (16 sections, 18 theorems, 166 equations)

This paper contains 16 sections, 18 theorems, 166 equations.

Table of Contents

  1. Introduction
  2. Notation and conventions
  3. Measure estimate of a certain family of sets
  4. Recap of the lower bound in (\ref{['equ:meestksy']})
  5. Proof of Theorem \ref{['thm:meaestss']}
  6. Some preparations for the proof
  7. Equidistribution and doubly mixing of certain $g_s$-translates
  8. Reduction to dynamics
  9. Measure estimates and smooth approximations
  10. Proof of Theorem \ref{['thm:improvedirichleconwei']}
  11. Convergence case
  12. Divergence case
  13. A reduction lemma
  14. The new ingredient: a disjointness fact
  15. Applications of effective equidistribution and double mixing theorems
  16. ...and 1 more sections

Key Result

Theorem 1

For any $A\in \mathrm{M}_{m,n}(\mathbb{R})$ and $t>1$, there exists $(\bm{p},\bm{q})\in \mathbb{Z}^m\times (\mathbb{Z}^n\setminus\{\bm{0}\})$ satisfying the following system of inequalities: Here $\|\cdot\|$ denotes the supremum norm on $\mathbb{R}^m$ and $\mathbb{R}^n$ respectively.

Theorems & Definitions (35)

  • Theorem 1
  • Theorem 2: KleinbockStrombergssonYu2022
  • Theorem 3
  • Remark 1.11
  • Remark 1.14
  • Theorem 2.1: KleinbockStrombergssonYu2022
  • Theorem 2.2
  • Proposition 2.3
  • Proposition 2.4
  • proof
  • ...and 25 more