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On dichotomy law for beta-dynamical system in parameter space

  • Fan LüEmail author
  • Jun Wu
Article
  • 16 Downloads

Abstract

Let \(\varphi :\mathbb {N}\rightarrow (0,1]\) be a positive function and \(T_{\beta }\) be the beta-transformation for any \(\beta >1\). We prove that the set
$$\begin{aligned} E(0, \varphi )=\{\beta >1:|T^{n}_{\beta }1-0|<\varphi (n) \text { for infinitely many } n\in \mathbb {N}\} \end{aligned}$$
is of zero or full Lebesgue measure in \((1,+\infty )\) according to \(\sum \varphi (n)<+\infty \) or not. As an application, we determine the exact Lebesgue measure of the set
$$\begin{aligned} \mathfrak {E}(0, \{l_{n}\})=\{\beta >1:|T^{n}_{\beta }1-0|<\beta ^{-l_{n}} \text { for infinitely many } n\in \mathbb {N}\}, \end{aligned}$$
where \(\{l_{n}\}_{n\ge 1}\) is a sequence of non-negative real numbers.

Keywords

Beta-dynamical system Diophantine approximation Shrinking target problem Paley–Zygmund inequality Lebesgue measure 

Mathematics Subject Classification

Primary 11K55 Secondary 28A80 

Notes

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsSichuan Normal UniversityChengduPeople’s Republic of China
  2. 2.School of Mathematics and StatisticsHuazhong University of Science and TechnologyWuhanPeople’s Republic of China

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