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

TL;DR

This paper studies the shrinking target property for irrational rotations on tori, generalizing previous results with a new approach to include weighted and $S$-arithmetic settings, enhancing understanding of approximation properties.

Contribution

It introduces a weighted effective analogue of the shrinking target property and extends results to the broader $S$-arithmetic setting, advancing the theory of irrational rotations.

Findings

Generalized shrinking target property to weighted settings

Extended results to $S$-arithmetic context

Provided new methods for effective approximation analysis

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.