Simultaneously dynamical Diophantine approximation in beta expansions

Authors

  • W. Wang Huazhong University of Science and Technology, Wuhan, China
  • L. Li Department of Mathematics, West Anhui University, Lu’an, Anhui 237012, China

Keywords:

Beta-expansions, Diophantine approximation, Hausdorff dimension

Abstract

Let $\beta>1$ be a real number and define the $\beta$-transformation on $[0,1]$ by $T_\beta:x\mapsto\beta x({\rm{mod}}\ 1).$ Let $f:[0,1]\rightarrow [0,1]$ and $g:[0,1]\rightarrow [0,1]$ be two Lipschitz functions. The main result of the paper is the Hausdorff dimension of the set \begin{align*}W( f,g, \tau_{1},\tau_{2})=\Big\{(x,y)\in [0,1]^2&:|T_{\beta}^{n}x-f(x)|<\beta^{-n\tau_1(x)},\\ |T_{\beta}^{n}y-&g(y)|< \beta^{-n\tau_{2}(y)} \text{~~~for infinitely many}\ n\in \N\Big\},\end{align*} where $\tau_1,\tau_2$ are two positive continuous functions with $\tau_1(x)\leq \tau_2(y)$ for all $x,y\in [0,1]$.

Published

2020-07-31

Issue

Vol. 102 No. 2 (2020)

Section

Articles
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