The distribution of the large partial quotients in continued fraction expansions  

作　　者：Bo Tan Chen Tian Baowei Wang 

机构地区：[1]School of Mathematics and Statistics,Huazhong University of Science and Technology,Wuhan 430074,China

出　　处：《Science China Mathematics》2023年第5期935-956,共22页中国科学：数学（英文版）

基　　金：supported by National Natural Science Foundation of China(Grant Nos.12171172 and 11831007)。

摘　　要：The existence of large partial quotients destroys many limit theorems in the metric theory of continued fractions.To achieve some variant forms of limit theorems,a common approach mostly used in practice is to discard the largest partial quotient,while this approach works in obtaining limit theorems only when there cannot exist two terms of large partial quotients in a metric sense.Motivated by this,we are led to consider the metric theory of points with at least two large partial quotients.More precisely,denoting by[a1(x),a2(x),...]the continued fraction expansion of x∈[0,1)and lettingψ:N→R+be a positive function tending to in nity as n→∞,we present a complete characterization on the metric properties of the set,i.e.,E(ψ)={x∈[0,1):∃16 k̸=ℓ6 n,ak(x)>ψ(n),aℓ(x)>ψ(n)for in nitely many n∈N}in the sense of the Lebesgue measure(the Borel-Bernstein type result)and the Hausdor dimension(the Jarnik type result).The main result implies that any nite deletion from a1(x)+……+an(x)cannot result in a law of large numbers.

关 键 词：continued fraction Hausdor dimension Borel-Bernstein theorem 

分 类 号：O211.3[理学—概率论与数理统计]