MULTIFRACTAL ANALYSIS OF CONVERGENCE EXPONENTS FOR PRODUCTS OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS

doi:10.1007/s10473-024-0422-6

数学物理学报(英文版) ›› 2024, Vol. 44 ›› Issue (4): 1594-1608.doi: 10.1007/s10473-024-0422-6

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MULTIFRACTAL ANALYSIS OF CONVERGENCE EXPONENTS FOR PRODUCTS OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS

Lulu Fang¹, Jihua Ma², Kunkun Song^3,*, Xin Yang³

1. School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, China;
2. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China;
3. Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China

收稿日期:2023-04-10 修回日期:2023-10-21 出版日期:2024-08-25 发布日期:2024-08-30

MULTIFRACTAL ANALYSIS OF CONVERGENCE EXPONENTS FOR PRODUCTS OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS

Lulu Fang¹, Jihua Ma², Kunkun Song^3,*, Xin Yang³

1. School of Mathematics and Statistics, Nanjing University of Science and Technology, Nanjing 210094, China;
2. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China;
3. Key Laboratory of Computing and Stochastic Mathematics (Ministry of Education), School of Mathematics and Statistics, Hunan Normal University, Changsha 410081, China

Received:2023-04-10 Revised:2023-10-21 Online:2024-08-25 Published:2024-08-30
Contact: *E-mail: songkunkun@hunnu.edu.cn
About author:E-mail: fanglulu1230@gmail.com; jhma@whu.edu.cn; xyang567@163.com
Supported by:
The research was supported by the Scientific Research Fund of Hunan Provincial Education Department (21B0070), the Natural Science Foundation of Jiangsu Province (BK20231452), the Fundamental Research Funds for the Central Universities (30922010809) and the National Natural Science Foundation of China (11801591, 11971195, 12071171, 12171107, 12201207, 12371072).

摘要/Abstract

摘要：

For each real number $x \in (0,1)$, let $[a_1(x),a_2(x),\cdots , a_n(x),\cdots ]$ denote its continued fraction expansion. We study the convergence exponent defined by

$\tau(x):= \inf\Big\{s \geq 0: \sum\limits_{n=1}^{\infty}\big(a_n(x)a_{n+1}(x)\big)^{-s}<\infty\Big\},$

which reflects the growth rate of the product of two consecutive partial quotients. As a main result, the Hausdorff dimensions of the level sets of $\tau(x)$ are determined.

关键词: continued fractions, product of partial quotients, Hausdorff dimension

Abstract:

For each real number $x \in (0,1)$, let $[a_1(x),a_2(x),\cdots , a_n(x),\cdots ]$ denote its continued fraction expansion. We study the convergence exponent defined by

$\tau(x):= \inf\Big\{s \geq 0: \sum\limits_{n=1}^{\infty}\big(a_n(x)a_{n+1}(x)\big)^{-s}<\infty\Big\},$

which reflects the growth rate of the product of two consecutive partial quotients. As a main result, the Hausdorff dimensions of the level sets of $\tau(x)$ are determined.

Key words: continued fractions, product of partial quotients, Hausdorff dimension

中图分类号:

11K50

Lulu Fang, Jihua Ma, Kunkun Song, Xin Yang. MULTIFRACTAL ANALYSIS OF CONVERGENCE EXPONENTS FOR PRODUCTS OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS[J]. 数学物理学报(英文版), 2024, 44(4): 1594-1608.

Lulu Fang, Jihua Ma, Kunkun Song, Xin Yang. MULTIFRACTAL ANALYSIS OF CONVERGENCE EXPONENTS FOR PRODUCTS OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS[J]. Acta mathematica scientia,Series B, 2024, 44(4): 1594-1608.

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MULTIFRACTAL ANALYSIS OF CONVERGENCE EXPONENTS FOR PRODUCTS OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS

MULTIFRACTAL ANALYSIS OF CONVERGENCE EXPONENTS FOR PRODUCTS OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS

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