Lüroth展式中数字序列具有一定缺项性质的点集的维数  

Dimension of the Set with Lacunarity Property of the Partial Quotients in the Lüroth Expansions

作　　者：邓静 DENG Jing(College of Mathematical Science,Chongqing Normal University,Shapingba 401331,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》2025年第2期34-38,共5页Journal of Shangluo University

基　　金：重庆市教育委员会科学技术研究项目(KJQN202100528);重庆市自然科学基金面上项目(CSTB2022NSCQ-MSX1255)。

摘　　要：对任意的实数x∈(0,1],设x=[d_(1)(x),d_(2)(x),...,dn(x),...]是其Lüroth展式,主要研究对满足一定缺项性质的点集的分形维数进行刻画。由于该类集合是已知维数的集合的子集,上界为1/2是显然的,只需刻画该集合维数的下界。为得到精确的分形维数下界,通过构造合适的Cantor型子集,利用经典的质量分布原理,求出其Hausdorff维数的下界为1/2,从而确定数字序列具有不同缺项性质的点集的Hausdorff维数为1/2。该结果给出了实数的Lüroth展式中数字序列{d_(n)(x)}_(n≥1)更加精细的不同增长速度的刻画,丰富了Lüroth展式的性质研究。For any x∈(0,1],let x=[d_(1)(x),d_(2)(x),...,dn(x),...]be the Lüroth series,the fractal dimension of the sets of points with different lacunarity on their partial quotie nts is described.Since such sets are subsets of the set with known dimensions,it is obvious that the upper bound is 1/2,and only the lower bound of the set's dimension needs to be described.In order to obtain the exact lower bound of fractal dimension,by constructing a suitable subset of Cantor type and using the classical mass distribution principle,the lower bound of its Hausdorff dimension is obtained to be 1/2,so that the Hausdorff dimension is determined to be 1/2 for the sets of points with different lacunarity on their partial quotients.This result gives a m ore detailed description of the different growth rates of digits{d_(n)(x)}_(n≥1) in Lüroth expansion of real numbers,which enriches the property research of Lüroth expansion.

关 键 词：数字序列 Lüroth展式 HAUSDORFF维数 

分 类 号：O156[理学—数学]