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HAUSDORFF DIMENSION OF THE SET APPROXIMATED BY IRRATIONAL ROTATIONS

  • Dong Han Kim (a1), Michał Rams (a2) and Baowei Wang (a3)
Abstract

Let $\unicode[STIX]{x1D703}$ be an irrational number and $\unicode[STIX]{x1D711}:\mathbb{N}\rightarrow \mathbb{R}^{+}$ be a monotone decreasing function tending to zero. Let

$$\begin{eqnarray}E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})=\{y\in \mathbb{R}:\Vert n\unicode[STIX]{x1D703}-y\Vert <\unicode[STIX]{x1D711}(n),\text{for infinitely many }n\in \mathbb{N}\},\end{eqnarray}$$
i.e. the set of points which are approximated by the irrational rotation with respect to the error function $\unicode[STIX]{x1D711}(n)$ . In this article, we give a complete description of the Hausdorff dimension of $E_{\unicode[STIX]{x1D711}}(\unicode[STIX]{x1D703})$ for any monotone function $\unicode[STIX]{x1D711}$ and any irrational $\unicode[STIX]{x1D703}$ .

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Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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