Limit theorems related to beta-expansion and continued fraction expansion

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• 作者：Lulu Fang ^{fanglulu1230@163.com" class="auth_mail" title="E-mail the corresponding author} ; Min Wu ^{wumin@scut.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Bing Li ; ^{scbingli@scut.edu.cn" class="auth_mail" title="E-mail the corresponding author}
• 关键词：11K50 ; 11K55 ; 60F05 ; 60F15
• 刊名：Journal of Number Theory
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：163
• 期：Complete
• 页码：385-405
• 全文大小：399 K

文摘

Let β>1$\beta >1$ be a real number and x∈[0,1)$x\in [0,1)$ be an irrational number. Denote by k_n(x)$k_n(x)$ the exact number of partial quotients in the continued fraction expansion of x given by the first n digits in the β-expansion of x   (n∈N$n\in \mathbb{N}$). In this paper, we show a central limit theorem and a law of the iterated logarithm for the random variables sequence {k_n,n≥1}$\{k_n,n\ge 1\}$, which generalize the results of Faivre [8] and Wu [31] respectively from e23690ba4b96b37ec4bf0ef4272d" title="Click to view the MathML source">β=10$\beta =10$ to any β>1$\beta >1$.