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HAUSDORFF DIMENSION OF THE SET APPROXIMATED BY IRRATIONAL ROTATIONS

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Classical measure theory Diophantine approximation, transcendental number theory

Published online by Cambridge University Press:  14 February 2018

Dong Han Kim ,
Michał Rams  and
Baowei Wang
Dong Han Kim
Affiliation:
Department of Mathematics Education, Dongguk University – Seoul, Seoul 04620, Korea email [email protected]
Michał Rams
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, 00-956 Warszawa, Poland email [email protected]
Baowei Wang
Affiliation:
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China email [email protected]
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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 $\unicode[STIX]{x1D703}$.

MSC classification

Primary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Secondary: 11J71: Distribution modulo one 28A80: Fractals
Type
Research Article
Information
Mathematika , Volume 64 , Issue 1 , 2018 , pp. 267 - 283
Copyright
Copyright © University College London 2018 

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