数学物理学报 ›› 2022, Vol. 42 ›› Issue (4): 978-1002.

• 论文 • 上一篇    下一篇

β-变换中一致丢番图逼近问题的维数理论

吴万楼1(),郑丽璇2,*()   

  1. 1 江苏师范大学 数学与统计学院 江苏徐州 221116
    2 广东财经大学 统计与数学系 广州 510320
  • 收稿日期:2021-04-08 出版日期:2022-08-26 发布日期:2022-08-08
  • 通讯作者: 郑丽璇 E-mail:wuwanlou@163.com;lixuan.zheng@gdufe.edu.cn
  • 作者简介:吴万楼,E-mail: wuwanlou@163.com
  • 基金资助:
    国家自然科学基金(12001245);江苏省自然科学基金(BK20201025);广东省自然科学基金(2020A1515110910)

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)

摘要:

$T_{\beta}$ (其中$\beta>1$) 为定义在区间$[0, 1)$上的$\beta$ -变换. 该文研究了$T_{\beta}$中轨道具有一致丢番图逼近性质的点组成的集合的分形维数, 具体而言, 对两个给定的正函数$\psi_1, \ \psi_2:{\Bbb N}\rightarrow{\Bbb R}^+$, 定义 其中$\gg$表示足够大. 该文计算了集合${\cal L}(\psi_1)\cap{\cal U}(\psi_2)$的豪斯道夫维数. 作为推论, 该文还得到了集合${\cal U}(\psi_2)$的豪斯道夫维数. 该文将文献[4] 中的结果进行了一般化, 文献[4] 中的函数$\psi_1, \ \psi_2$仅仅是指数函数.

关键词: $\beta$ -变换, 一致丢番图逼近, 豪斯道夫维数

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

中图分类号: 

  • O211