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A NOTE ON INHOMOGENEOUS DIOPHANTINE APPROXIMATION IN BETA-DYNAMICAL SYSTEM

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Classical measure theory

Published online by Cambridge University Press:  12 September 2014

YUEHUA GE  and
FAN LÜ
YUEHUA GE
Affiliation:
Huazhong University of Science and Technology, Wuhan 430074, PR China email geyuehua1001@126.com
FAN LÜ*
Affiliation:
Huazhong University of Science and Technology, Wuhan 430074, PR China email lvfan1123@163.com
*
✉lvfan1123@163.com
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Abstract

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We study the distribution of the orbits of real numbers under the beta-transformation $T_{{\it\beta}}$ for any ${\it\beta}>1$. More precisely, for any real number ${\it\beta}>1$ and a positive function ${\it\varphi}:\mathbb{N}\rightarrow \mathbb{R}^{+}$, we determine the Lebesgue measure and the Hausdorff dimension of the following set:

$$\begin{eqnarray}E(T_{{\it\beta}},{\it\varphi})=\{(x,y)\in [0,1]\times [0,1]:|T_{{\it\beta}}^{n}x-y|<{\it\varphi}(n)\text{ for infinitely many }n\in \mathbb{N}\}.\end{eqnarray}$$

Keywords

beta-dynamical systembeta-expansioninhomogeneous Diophantine approximationLebesgue measureHausdorff dimension

MSC classification

Primary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Secondary: 28A80: Fractals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 1 , February 2015 , pp. 34 - 40
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

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