The distribution of the large partial quotients in continued fraction expansions

作     者：Bo Tan Chen Tian Baowei Wang 

作者机构：School of Mathematics and StatisticsHuazhong University of Science and TechnologyWuhan 430074China 

出 版 物：《Science China Mathematics》 (中国科学：数学（英文版）)

年 卷 期：2023年第66卷第5期

页      面：935-956页

核心收录：

学科分类：02[经济学] 0202[经济学-应用经济学] 020208[经济学-统计学] 07[理学] 0714[理学-统计学（可授理学、经济学学位）] 070103[理学-概率论与数理统计] 0701[理学-数学] 

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

主　　题：continued fraction Hausdor dimension Borel-Bernstein theorem 

摘      要：The existence of large partial quotients destroys many limit theorems in the metric theory of continued *** 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 *** by this,we are led to consider the metric theory of points with at least two large partial *** 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.