Univoque bases and Hausdorff dimension

被引：10

作者：

Kong, Derong ^{[1
,2
]}

Li, Wenxia ^{[3
]}

Lu, Fan ^{[4
]}

de Vries, Martijn ^{[5
]}

机构：

[1] Yangzhou Univ, Sch Math Sci, Yangzhou 225002, Jiangsu, Peoples R China

[2] Leiden Univ, Math Inst, POB 9512, NL-2300 RA Leiden, Netherlands

[3] East China Normal Univ, Dept Math, Shanghai Key Lab PMMP, Shanghai 200062, Peoples R China

[4] Sichuan Normal Univ, Dept Math, Chengdu 610068, Sichuan, Peoples R China

[5] Tussen Grachten 213, NL-1381 DZ Weesp, Netherlands

来源：

MONATSHEFTE FUR MATHEMATIK | 2017年 / 184卷 / 03期

关键词：

Univoque bases; Univoque sets; Hausdorff dimensions; Generalized Thue-Morse sequences; UNIQUE EXPANSIONS; SETS;

D O I：

10.1007/s00605-017-1047-9

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

Given a positive integer M and a real number , a q -expansion of a real number x is a sequence with such that x = Sigma(infinity)(i=1)c(i)q(-i) It is well known that if , then each has a q-expansion. Let be the set of univoque bases for which 1 has a unique q-expansion. The main object of this paper is to provide new characterizations of and to show that the Hausdorff dimension of the set of numbers with a unique q-expansion changes the most if q "crosses" a univoque base. Denote by the set of such that there exist numbers having precisely two distinct q-expansions. As a by-product of our results, we obtain an answer to a question of Sidorov (J Number Theory 129:741-754, 2009) and prove that dimH(B-2 boolean AND (q ', q ' + delta)) > 0 for any delta > 0, where is the Komornik-Loreti constant.

引用

页码：443 / 458

页数：16

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