GROWTH RATE OF DIGITS IN A GAUSS-LIKE IFS

doi:10.1007/s10473-026-0117-2

数学物理学报(英文版) ›› 2026, Vol. 46 ›› Issue (1): 293-310.doi: 10.1007/s10473-026-0117-2

GROWTH RATE OF DIGITS IN A GAUSS-LIKE IFS

Saisai SHI¹, Bo TAN^2,*, Qinglong ZHOU³

1. Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233030, China;
2. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China;
3. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

收稿日期:2024-09-23 修回日期:2024-12-06 出版日期:2026-01-25 发布日期:2026-05-22

GROWTH RATE OF DIGITS IN A GAUSS-LIKE IFS

Saisai SHI¹, Bo TAN^2,*, Qinglong ZHOU³

1. Institute of Statistics and Applied Mathematics, Anhui University of Finance and Economics, Bengbu 233030, China;
2. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China;
3. School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

Received:2024-09-23 Revised:2024-12-06 Online:2026-01-25 Published:2026-05-22
Contact: * Bo TAN, E-mail: tanbo@hust.edu.cn
About author:Saisai SHI,E-mail: saisai shi@126.com;Qinglong ZHOU, E-mail: zhouql@whut.edu.cn
Supported by:
Scientific Research Project of Colleges and Universities in Anhui Province (2024AH050016). The second author was supported by the NSFC (12171172). The third author was supported by the NSFC (12201476) and the Fundamental Research Funds for the Central Universities.

摘要/Abstract

摘要： Let $\Psi=\{\psi_{n}\}_{n\geq1}$ be an iterated function system (IFS) on [0,1] with attractor $J.$ Associated with each $x\in J,$ there is a sequence $\{\omega_{n}(x)\}_{n\geq 1}$ consisting of integers, called the digit sequence of $x,$ such that
$x=\lim_{n\rightarrow\infty}\psi_{\omega_{1}(x)}\circ\cdots\circ \psi_{\omega_{n}(x)}$(1).
We revisit the Borel-Bernstein theorem in a $d$-decaying Gauss-like IFS, and completely characterize the metrical properties of the set
$E(\Phi)=\big\{x\in J\colon \omega_{n}(x)\geq \Phi(n) \text{ for infinitely many } n\in \mathbb{N}\big\},$
where $\Phi\colon \mathbb{N}\rightarrow \mathbb{R}$ is a positive function.

关键词: Gauss-like iterated function system, Diophantine approximation, Hausdorff dimension

Abstract: Let $\Psi=\{\psi_{n}\}_{n\geq1}$ be an iterated function system (IFS) on [0,1] with attractor $J.$ Associated with each $x\in J,$ there is a sequence $\{\omega_{n}(x)\}_{n\geq 1}$ consisting of integers, called the digit sequence of $x,$ such that
$x=\lim_{n\rightarrow\infty}\psi_{\omega_{1}(x)}\circ\cdots\circ \psi_{\omega_{n}(x)}$(1).
We revisit the Borel-Bernstein theorem in a $d$-decaying Gauss-like IFS, and completely characterize the metrical properties of the set
$E(\Phi)=\big\{x\in J\colon \omega_{n}(x)\geq \Phi(n) \text{ for infinitely many } n\in \mathbb{N}\big\},$
where $\Phi\colon \mathbb{N}\rightarrow \mathbb{R}$ is a positive function.

Key words: Gauss-like iterated function system, Diophantine approximation, Hausdorff dimension

中图分类号:

28A80

Saisai SHI¹, Bo TAN^2,*, Qinglong ZHOU³. GROWTH RATE OF DIGITS IN A GAUSS-LIKE IFS[J]. 数学物理学报(英文版), 2026, 46(1): 293-310.

Saisai SHI¹, Bo TAN^2,*, Qinglong ZHOU³. GROWTH RATE OF DIGITS IN A GAUSS-LIKE IFS[J]. Acta mathematica scientia,Series B, 2026, 46(1): 293-310.

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GROWTH RATE OF DIGITS IN A GAUSS-LIKE IFS

GROWTH RATE OF DIGITS IN A GAUSS-LIKE IFS

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