Acta mathematica scientia,Series A ›› 2022, Vol. 42 ›› Issue (4): 978-1002.

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Dimension Theory of Uniform Diophantine Approximation Related to Beta-Transformations

Wanlou Wu1(),Lixuan Zheng2,*()   

  1. 1 School of Mathematics and Statistics, Jiangsu Normal University, Jiangsu Xuzhou 221116
    2 Department of Statistics and Mathematics, Guangdong University of Finance and Economics, Guangzhou 510320
  • Received:2021-04-08 Online:2022-08-26 Published:2022-08-08
  • Contact: Lixuan Zheng E-mail:wuwanlou@163.com;lixuan.zheng@gdufe.edu.cn
  • Supported by:
    the NSFC(12001245);the NSF of Jiangsu Province(BK20201025);the NSF of Guangdong Province(2020A1515110910)

Abstract:

For $\beta>1$, let $T_\beta$ be the $\beta$-transformation defined on $[0, 1)$. We study the sets of points whose orbits of $T_\beta$ have uniform Diophantine approximation properties. Precisely, for two given positive functions $\psi_1, \ \psi_2:{\Bbb N}\rightarrow{\Bbb R}^+$, define where $\gg$ means large enough. We calculate the Hausdorff dimension of the set ${\cal L}(\psi_1)\cap{\cal U}(\psi_2)$. As a corollary, we obtain the Hausdorff dimension of the set ${\cal U}(\psi_2)$. Our work generalizes the results of [4] where only exponential functions $\psi_1, \ \psi_2$ were taken into consideration.

Key words: Beta-transformation, Uniform Diophantine approximation, Hausdorff dimension

CLC Number: 

  • O211
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