On the refined shrinking target property of rotations

TL;DR

This paper investigates the conditions under which irrational rotations exhibit the shrinking target property, providing new criteria for when the lim inf of scaled distances approaches zero for almost every point.

Contribution

It establishes precise conditions on irrational numbers and functions for the shrinking target property to hold in rotations, advancing understanding of dynamical recurrence behaviors.

Findings

Identifies conditions on heta and $ \

Characterizes irrationals with universal shrinking target property for divergent series \

Abstract

We discuss the shrinking target property of irrational rotations. We obtain the condition of an irrational $\theta$ and monotone increasing $\varphi(n)$ such that $$ \liminf_{n \to \infty} n \varphi (n) \| n\theta - s \| = 0 \text{ for almost every } s. $$ We also consider the class of irrationals for which the limit inferior is 0 for every monotone increasing $\varphi(n)$ such that $\sum_n 1/(n\varphi(n))$ diverges.