Engel连分数展式中数字序列的增长速度  

The Growth Rate of Number Sequence in Engel Continued Fraction

作　　者：胡林 HU Lin(College of Mathematical Science,Chongqing Normal University,Shapingba401331,Chongqing)

机构地区：[1]重庆师范大学数学科学学院,重庆沙坪坝401331

出　　处：《商洛学院学报》2024年第6期28-34,共7页Journal of Shangluo University

基　　金：重庆市教育委员会科学技术研究项目(KJQN202100528)。

摘　　要：对任意的实数x∈[0,1],设x=[b_(1)(x),b_(2)(x),…]为其Engel连分数展式。为探究Engel连分数展式中数字序列的增长速度问题。研究了Engel连分数展式中lnb_n(x)以线性速度增长时相关例外集E(α,β)={x∈[0,1):liminf_(n→∞)lnbn(x)/n=α,limsup_(n→∞)lnb_(n)(x)/n=β}的Hausdorff维数,通过构建例外集的Cantor子集并利用质量分布原理,给出了对任意的0≤α≤β≤∞,例外集E(α,β)都是满维的结果,该结果补充了Engel连分数展式中数字序列增长速度问题的研究。For any real number x∈[0,1),let x=[b1(x),b2(x),…]be its Engel continued fraction.To explore the growth rate of number sequence in Engel continued fraction expansion,this study mainly describes the Hausdorff dimension of the relevant exception set E(α,β)={x∈[0,1):liminf_(n→∞)lnbn(x)/n=α,limsup_(n→∞)lnb_(n)(x)/n=β}when lnbn(x)in Engel continued fractional expansion grows at a linear rate.By constructing the Cantor subset of the exception set and using the mass distribution principle,the result that the exception set E(α,β)is full dimension for any 0≤α≤β≤∞is given.This result complements the research on the growth rate of number sequences in Engel continued fraction expansion.

关 键 词：Engel连分数展式 HAUSDORFF维数 例外集 

分 类 号：O156[理学—数学]