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

doi:10.1007/s10473-024-0422-6

Acta mathematica scientia,Series B ›› 2024, Vol. 44 ›› Issue (4): 1594-1608.doi: 10.1007/s10473-024-0422-6

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).

Abstract

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

CLC Number:

11K50

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.

Trendmd

References

[1] Bakhtawar A, Feng J. Increasing rate of weighted product of partial quotients in continued fractions. Chaos Solitons Fractals, 2023, 172: Art 113591
[2] Bakhtawar A, Bos P, Hussain M. Hausdorff dimension of an exceptional set in the theory of continued fractions. Nonlinearity, 2020, 33: 2615-2639
[3] Bakhtawar A, Bos P, Hussain M. The sets of Dirichlet non-improvable numbers versus well-approximable numbers. Ergodic Theory Dynam Systems, 2020, 40: 3217-3235
[4] Bos P, Hussain M, Simmons D. The generalised Hausdorff measure of sets of Dirichlet non-improvable numbers. Proc Amer Math Soc, 2023, 151: 1823-1838
[5] Bosma W, Dajani K, Kraaikamp C. Entropy quotients and correct digits in number-theoretic expansions. IMS Lecture Notes Monogr Ser: Dynamics & Stochastics, 2006, 48: 176-188
[6] Davenport H, Schmidt W.Dirichlet's theorem on diophantine approximation//Symposia Mathematica. London: Academic Press, 1970: 113-132
[7] Falconer K.Techniques in Fractal Geometry. New York: Wiley, 1997
[8] Fan A, Liao L, Wang B, Wu J. On Khintchine exponents and Lyapunov exponents of continued fractions. Ergodic Theory Dynam Systems, 2009, 29: 73-109
[9] Fang L, Ma J, Song K, Wu M.Multifractal analysis of the convergence exponent in continued fractions. Acta Math Sci, 2021, 41B: 1896-1910
[10] Feng J, Xu J. Sets of Dirichlet non-improvable numbers with certain order in the theory of continued fractions. Nonlinearity, 2021, 34: 1598-1611
[11] Good I. The fractional dimensional theory of continued fractions. Proc Cambridge Philos Soc, 1941, 37: 199-228
[12] Hu H, Hussain M, Yu Y. Limit theorems for sums of products of consecutive partial quotients of continued fractions. Nonlinearity, 2021, 34: 8143-8173
[13] Huang L, Wu J. Uniformly non-improvable Dirichlet set via continued fractions. Proc Amer Math Soc, 2019, 147: 4617-4624
[14] Huang L, Wu J, Xu J. Metric properties of the product of consecutive partial quotients in continued fractions. Israel J Math, 2020, 238: 901-943
[15] Hussain M, Kleinbock D, Wadleigh N, Wang B. Hausdorff measure of sets of Dirichlet non-improvable numbers. Mathematika, 2018, 64: 502-518
[16] Iosifescu M, Kraaikamp C.Metrical Theory of Continued Fractions. Dordrecht: Kluwer Academic Publishers, 2002
[17] Jaffard S, Martin B. Multifractal analysis of the Brjuno function. Invent Math, 2018, 212: 109-132
[18] Jarník V. Zur metrischen Theorie der diopahantischen Approximationen. Proc Mat Fyz, 1928, 36: 91-106
[19] Kesseböhmer M, Stratmann B. Fractal analysis for sets of non-differentiability of Minkowski's question mark function. J Number Theory, 2008, 128: 2663-2686
[20] Khintchine A. Continued Fractions.Chicago: The University of Chicago Press, 1964
[21] Kleinbock D, Wadleigh N. A zero-one law for improvements to Dirichlet's Theorem. Proc Amer Math Soc, 2018, 146: 1833-1844
[22] Nicolay S, Simons L. About the multifractal nature of Cantor's bijection: bounds for the Hölder exponent at almost every irrational point. Fractals, 2016, 24: Art 1650014
[23] Pollicott M, Weiss H. Multifractal analysis of Lyapunov exponent for continued fraction and Manneville-Pomeau transformations and applications to Diophantine approximation. Comm Math Phys, 1999, 207: 145-171
[24] Pólya G, Szegö G.Problems and Theorems in Analysis I: Series, Integral Calculus, Theory of Functions. Berlin: Springer-Verlag, 1972
[25] Ramharter G. Eine Bemerkungüber gewisse Nullmengen von Kettenbrüchen. Ann Univ Sci Budapest Eötvös Sect Math, 1985, 28: 11-15
[26] Zhang L. Set of extremely Dirichlet non-improvable points. Fractals, 2020, 28: Art 2050034

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

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 10

[1]	Jia LIU, Saisai SHI, Yuan ZHANG. ON THE GRAPHS OF PRODUCTS OF CONTINUOUS FUNCTIONS AND FRACTAL DIMENSIONS^* [J]. Acta mathematica scientia,Series B, 2023, 43(6): 2483-2492.
[2]	Qinghui LIU, Zhiyi Tang. THE HAUSDORFF DIMENSION OF THE SPECTRUM OF A CLASS OF GENERALIZED THUE-MORSE HAMILTONIANS^* [J]. Acta mathematica scientia,Series B, 2023, 43(5): 1997-2004.
[3]	Yajuan ZHAO, Yongsheng LI, Wei YAN, Xiangqian YAN. ON THE DIMENSION OF THE DIVERGENCE SET OF THE OSTROVSKY EQUATION [J]. Acta mathematica scientia,Series B, 2022, 42(4): 1607-1620.
[4]	Jun WANG, Zhenlong CHEN. HITTING PROBABILITIES AND INTERSECTIONS OF TIME-SPACE ANISOTROPIC RANDOM FIELDS [J]. Acta mathematica scientia,Series B, 2022, 42(2): 653-670.
[5]	Lulu FANG, Jihua MA, Kunkun SONG, Min WU. MULTIFRACTAL ANALYSIS OF THE CONVERGENCE EXPONENT IN CONTINUED FRACTIONS [J]. Acta mathematica scientia,Series B, 2021, 41(6): 1896-1910.
[6]	Changgui ZHANG. ON THETA-TYPE FUNCTIONS IN THE FORM (x;q)_∞ [J]. Acta mathematica scientia,Series B, 2021, 41(6): 2086-2106.
[7]	Dan LI, Junfeng LI, Jie XIAO. AN UPBOUND OF HAUSDORFF'S DIMENSION OF THE DIVERGENCE SET OF THE FRACTIONAL SCHRÖDINGER OPERATOR ON H^s($\mathbb{R}^n$) [J]. Acta mathematica scientia,Series B, 2021, 41(4): 1223-1249.
[8]	Jianfeng ZHOU, Zhong TAN. REGULARITY OF WEAK SOLUTIONS TO A CLASS OF NONLINEAR PROBLEM [J]. Acta mathematica scientia,Series B, 2021, 41(4): 1333-1365.
[9]	Zhenliang ZHANG, Chunyun CAO. ON POINTS CONTAIN ARITHMETIC PROGRESSIONS IN THEIR LÜROTH EXPANSION [J]. Acta mathematica scientia,Series B, 2016, 36(1): 257-264.
[10]	CHEN Zhen-Long. ON INTERSECTIONS OF INDEPENDENT NONDEGENERATE DIFFUSION PROCESSES [J]. Acta mathematica scientia,Series B, 2014, 34(1): 141-161.
[11]	HU Dong-Gang, HU Xue-Hai. ARBITRARILY LONG ARITHMETIC PROGRESSIONS FOR CONTINUED FRACTIONS OF LAURENT SERIES [J]. Acta mathematica scientia,Series B, 2013, 33(4): 943-949.
[12]	CHEN Zhen-Long. FRACTAL PROPERTIES OF POLAR SETS OF RANDOM STRING PROCESSES [J]. Acta mathematica scientia,Series B, 2011, 31(3): 969-992.
[13]	CHEN Zhen-Long, LI Hui-Qiong. POLAR SETS OF MULTIPARAMETER BIFRACTIONAL BROWNIAN SHEETS [J]. Acta mathematica scientia,Series B, 2010, 30(3): 857-872.
[14]	Chen Zhenlong. INTERSECTIONS AND POLAR FUNCTIONS OF FRACTIONAL BROWNIAN SHEETS [J]. Acta mathematica scientia,Series B, 2008, 28(4): 779-796.
[15]	Gao Junyang; Qiao Jianyong. CONTINUITY AND HAUSDORFF DIMENSION OF JULIA SET CONCERNING YANG-LEE ZEROS [J]. Acta mathematica scientia,Series B, 2008, 28(3): 530-536.