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arXiv · 1609.08724

Hausdorff dimension of the set approximated by irrational rotations

Dong Han Kim·Michał Rams·Baowei Wang
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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 $\theta$.

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Dong Han Kim, Michał Rams, Baowei Wang. 2016-09-28. Hausdorff dimension of the set approximated by irrational rotations. https://doi.org/10.1112/s0025579317000523

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