数学物理学报 ›› 2024, Vol. 44 ›› Issue (4): 847-858.

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Lüroth展式中最大数字例外集的Hausdorff维数

陈俊攸(),张振亮*()   

  1. 重庆师范大学数学科学学院 重庆 401331
  • 收稿日期:2023-07-18 修回日期:2024-02-25 出版日期:2024-08-26 发布日期:2024-07-26
  • 通讯作者: *张振亮, E-mail:zhliang_zhang@163.com.
  • 作者简介:陈俊攸, E-mail:ll_chen2602@163.com
  • 基金资助:
    重庆市教育委员会科学技术研究项目(KJQN202100528);重庆市自然科学基金(CSTB2022NSCQMSX-1255)

On Hausdorff Dimension of the Exceptional Sets of Partial Maximal Digits for Lüroth Expansion

Chen Junyou(),Zhang Zhenliang*()   

  1. School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331
  • Received:2023-07-18 Revised:2024-02-25 Online:2024-08-26 Published:2024-07-26
  • Supported by:
    Science and Technology Research Program of Chongqing Municipal Education Commission(KJQN202100528);Natural Science Foundation of Chongqing(CSTB2022NSCQMSX-1255)

摘要:

对任意的$x\in (0,1)$,记 $x$ 的 Lüroth 展式为 $x=[ d_{1}(x), d_{2} (x), \cdots, d_{n} (x), \cdots ]$.记前$n$个数字中的最大数字为 $L_{n}(x)=\max \left\{d_{1}(x), \cdots, d_{n}(x)\right\},$对任意实数 $0< \alpha < \beta < \infty$,该文刻画了集合

$\begin{align*}F_{\phi}(\alpha, \beta)=\left\{x \in(0,1): \liminf _{n \rightarrow \infty} \frac{L_{n}(x)}{\phi(n)}=\alpha, \limsup _{n \rightarrow \infty} \frac{L_{n}(x)}{\phi(n)}=\beta\right\},\end{align*}$

的Hausdorff维数, 其中 $\phi (n)= n^{\gamma} (\gamma>0) $ 或者${\rm e}^{n^{\gamma}} (\gamma>0 )$. 该结论补充了文献[13]的结果. 类似地, 该文还考虑了Lüroth展式中数字部分和的相应例外集的Hausdorff维数.

关键词: Lüroth 展式, 最大数字, Hausdorff 维数

Abstract:

For any $ x\in (0,1)$, let $x=[ d_{1}(x), d_{2} (x), \cdots, d_{n} (x)]$ be its Lüroth expansion. Denote the maximal digits of the first $n$ digits by $L_{n}(x)=\max \left\{d_{1}(x), \cdots, d_{n}(x)\right\}.$ For any real number $0< \alpha < \beta < \infty$, we determine the Hausdorff dimension of the exceptional set

$F_{\phi}(\alpha, \beta)=\left\{x \in(0,1): \liminf _{n \rightarrow \infty} \frac{L_{n}(x)}{\phi(n)}=\alpha, \limsup _{n \rightarrow \infty} \frac{L_{n}(x)}{\phi(n)}=\beta\right\},$

where $ \phi (n)= n^{\gamma} (\gamma>0)$ or $ {\rm e}^{n^{\gamma}} (\gamma>0 )$. This supplements the results of [13]. Similarly, the corresponding exceptional sets of the sums of digits in Lüroth expansion are also studied.

Key words: Lüroth expansion, Largest digit, Hausdorff dimension

中图分类号: 

  • O156.7