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Rectangular shrinking targets for $\mathbb{Z}^m$ actions on tori: well and badly approximable systems

Victor Beresnevich, Shreyasi Datta, Anish Ghosh, Benjamin Ward

TL;DR

This paper advances the shrinking target problem for irrational rotations and $\mathbb{Z}^m$-actions on tori by introducing rectangular targets weighted by $\bm\tau$ and $\bm\eta$, and by extending the theory to the $S$-arithmetic setting. It develops a weighted analogue of Dirichlet's theorem and introduces the notion of $(\bm\tau,\bm\eta)$-non-singular matrices, proving metric, inhomogeneous approximation results for almost every target vector $\bm\gamma$. The authors combine lattice-geometry arguments (Minkowski's theorems), strong approximation, and measure-theoretic tools from BDGW_null to establish a twisted weighted asymptotic Dirichlet principle and its $S$-arithmetic generalization. The results generalize Kim–Shapira's and Shapira's theorems to weighted rectangular targets and broader arithmetic settings, enabling new insights into badly approximable systems and metric Diophantine approximation in homogeneous dynamics. This work thus broadens the scope of shrinking targets in dynamical systems and provides robust techniques for analyzing weighted, arithmetic, and inhomogeneous cases.

Abstract

In this paper we investigate the shrinking target property for irrational rotations. This was first studied by Kurzweil (1951) and has received considerable interest of late. Using a new approach, we generalize results of Kim (2007) and Shapira (2013) by proving a weighted effective analogue of the shrinking target property. Furthermore, our results are established in the much wider $S$-arithmetic setting.

Rectangular shrinking targets for $\mathbb{Z}^m$ actions on tori: well and badly approximable systems

TL;DR

This paper advances the shrinking target problem for irrational rotations and -actions on tori by introducing rectangular targets weighted by and , and by extending the theory to the -arithmetic setting. It develops a weighted analogue of Dirichlet's theorem and introduces the notion of -non-singular matrices, proving metric, inhomogeneous approximation results for almost every target vector . The authors combine lattice-geometry arguments (Minkowski's theorems), strong approximation, and measure-theoretic tools from BDGW_null to establish a twisted weighted asymptotic Dirichlet principle and its -arithmetic generalization. The results generalize Kim–Shapira's and Shapira's theorems to weighted rectangular targets and broader arithmetic settings, enabling new insights into badly approximable systems and metric Diophantine approximation in homogeneous dynamics. This work thus broadens the scope of shrinking targets in dynamical systems and provides robust techniques for analyzing weighted, arithmetic, and inhomogeneous cases.

Abstract

In this paper we investigate the shrinking target property for irrational rotations. This was first studied by Kurzweil (1951) and has received considerable interest of late. Using a new approach, we generalize results of Kim (2007) and Shapira (2013) by proving a weighted effective analogue of the shrinking target property. Furthermore, our results are established in the much wider -arithmetic setting.
Paper Structure (9 sections, 6 theorems, 82 equations)

This paper contains 9 sections, 6 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Real tori: a special case
  3. $S$-arithmetic setup and the main theorem
  4. A sketch of a proof of Theorem \ref{['thm1']}
  5. An auxiliary result
  6. Proof of Theorem \ref{['thm: S main theorem']}
  7. Proof of Theorem \ref{['thm: nonsingular Dirichlet coro']} in the case $\infty\notin S$
  8. Proof of Theorem \ref{['thm: nonsingular Dirichlet coro']} in the case $\infty \in S$
  9. Proof of Theorem \ref{['thm: S main theorem']} using Theorem \ref{['thm: nonsingular Dirichlet coro']}

Key Result

Theorem 1.1

Let $\theta$ be any irrational number. For almost every $s \in \mathbb{R}$ we have

Theorems & Definitions (11)

  • Theorem 1.1
  • Definition 1.2
  • Theorem 1.3
  • Theorem 2.1
  • Remark 2.2
  • Definition 2.3
  • Theorem 2.4
  • Remark 2.5
  • Lemma 4.1: BDGW_null Lemma 5.7
  • Theorem 5.1
  • ...and 1 more