Hausdorff dimension of the set approximated by irrational rotations

Abstract

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

15pages

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