A note on inhomogeneous Diophantine approximation in beta-dynamical system

Authors

  • Y. Ge
  • F. Lü

Keywords:

Beta-dynamical system, beta-expansion, inhomogeneous Diophantine approximation, Lebesgue measure, Hausdorff dimension

Abstract

We study the distribution of the orbits of real numbers under the beta-transformation \(T_{\beta}\) for any \(\beta>1\). More precisely, for any real number \(\beta>1\) and positive function \(\varphi: \mathbb{N}\rightarrow \mathbb{R}^{+}\), we determine the Lebesgue measure and the Hausdorff dimension of the following set \[E(T_{\beta},\varphi)=\{(x,y)\in[0,1]\times[0,1]: |T^{n}_{\beta}x-y|<\varphi(n) \text{ for infinitely many }n\in\mathbb{N}\}.\] DOI: 10.1017/S0004972714000677

Published

2014-11-13

Issue

Vol. 91 No. 1 (2015)

Section

Articles
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