β-变换中一致丢番图逼近问题的维数理论

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

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 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 Zheng^2,*()

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/Abstract

摘要：

令$T_{\beta}$ (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 ${\cal L}(\psi_1):=\left\{x\in[0, 1]:T_\beta^n x<\psi_1(n), \mbox{ 对无穷多个 $n\in{\Bbb N}$ 成立}\right\}, $ ${\cal U}(\psi_2):=\left\{x\in [0, 1]:\forall \ N\gg1, \ \exists \ n\in[0, N], \ s.t. \ T^n_\beta x<\psi_2(N)\right\}, $ 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 ${\cal L}(\psi_1):=\left\{x\in[0, 1]:T_\beta^n x<\psi_1(n), \mbox{ for infinitely many $n\in{\Bbb N}$}\right\}, $ ${\cal U}(\psi_2):=\left\{x\in [0, 1]:\forall \ N\gg1, \ \exists \ n\in[0, N], \ s.t. \ T^n_\beta x<\psi_2(N)\right\}, $ 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

吴万楼,郑丽璇. β-变换中一致丢番图逼近问题的维数理论[J]. 数学物理学报, 2022, 42(4): 978-1002.

Wanlou Wu,Lixuan Zheng. Dimension Theory of Uniform Diophantine Approximation Related to Beta-Transformations[J]. Acta mathematica scientia,Series A, 2022, 42(4): 978-1002.

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