Hausdorff dimension of the set approximated by irrational rotations

TL;DR

This paper characterizes the Hausdorff dimension of sets of points approximated by irrational rotations with a given error function, providing a comprehensive description for all monotone functions and irrational numbers.

Contribution

It offers a complete formula for the Hausdorff dimension of approximation sets related to irrational rotations for any monotone error function.

Findings

Explicit Hausdorff dimension formulas derived

Results apply to all monotone decreasing functions

Provides a unified framework for approximation by irrational rotations

Abstract

Let $\theta$ be an irrational number and $\varphi: {\mathbb N} \to {\mathbb R}^{+}$ be a monotone decreasing function tending to zero. Let $$E_\varphi(\theta) =\Big\{y \in \mathbb R: \|n\theta- y\|<\varphi(n), \ {\text{for infinitely many}}\ n\in {\mathbb N} \Big\}, $$ i.e. the set of points which are approximated by the irrational rotation with respect to the error function $\varphi(n)$. In this article, we give a complete description of the Hausdorff dimension of $E_\varphi(\theta)$ for any monotone function $\varphi$ and any irrational $\theta$.