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ON LÜROTH EXPANSIONS IN WHICH THE LARGEST DIGIT GROWS WITH SLOWLY INCREASING SPEED

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Classical measure theory

Published online by Cambridge University Press:  23 June 2022

MENGJIE ZHANG Open the ORCID record for MENGJIE ZHANG [Opens in a new window]  and
WEILIANG WANG Open the ORCID record for WEILIANG WANG [Opens in a new window]
MENGJIE ZHANG*
Affiliation:
School of Mathematics and Statistics, Henan University of Science and Technology, 471023 Luoyang, PR China
WEILIANG WANG
Affiliation:
School of Finance and Mathematics, West Anhui University, 237012 Luan, PR China e-mail: weiliang_wang@hust.edu.cn
*
e-mail: mengjie_zhang@haust.edu.cn
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Abstract

Let $0\leq \alpha \leq \infty $ , $0\leq a\leq b\leq \infty $ and $\psi $ be a positive function defined on $(0,\infty )$ . This paper is concerned with the growth of $L_{n}(x)$ , the largest digit of the first n terms in the Lüroth expansion of $x\in (0,1]$ . Under some suitable assumptions on the function $\psi $ , we completely determine the Hausdorff dimensions of the sets

$$\begin{align*}E_\psi(\alpha)=\bigg\{x\in(0,1]: \lim\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=\alpha\bigg\} \end{align*}$$

and

$$\begin{align*}E_\psi(a,b)=\bigg\{x\in(0,1]: \liminf\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=a, \limsup\limits_{n\rightarrow\infty}\frac{\log L_n(x)}{\log\psi(n)}=b\bigg\}. \end{align*}$$

Keywords

Lüroth expansionlargest digitHausdorff dimension

MSC classification

Primary: 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension
Secondary: 28A80: Fractals
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 107 , Issue 2 , April 2023 , pp. 204 - 214
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Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of Australian Mathematical Publishing Association Inc.

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Footnotes

This research was supported by National Natural Science Foundation of China (No. 12101191), Natural Science Research Project of West Anhui University (No. WGKQ2021020) and Provincial Natural Science Research Project of Anhui Colleges (No. KJ2021A0950).

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