Hausdorff dimension of the set approximated 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 $\theta$.