Hausdorff dimension of unique beta expansions

• Published in: Nonlinearity Vol. 28; no. 1; pp. 187 - 209
• Main Authors: Kong, Derong, Li, Wenxia
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.01.2015
• Subjects: admissible block; admissible interval; Fluctuation; generalized Thue-Morse sequence; Hausdorff dimension; Integers; Intervals; Mathematical analysis; Nonlinearity; Real numbers; Sequences; Texts; transcendental number; unique beta expansion
• ISSN: 0951-7715; 1361-6544
• DOI: 10.1088/0951-7715/28/1/187

Given an integer N 2 and a real number β > 1, let Γβ, N be the set of all with di ∈ {0, 1, ···, N − 1} for all i 1. The infinite sequence (di) is called a β-expansion of x. Let Uβ,N be the set of all x's in Γβ,N which have unique β-expansions. We give explicit formula of the Hausdorff dimension of Uβ,N for β in any admissible interval [βL, βU], where βL is a purely Parry number while βU is a transcendental number whose quasi-greedy expansion of 1 is related to the classical Thue-Morse sequence. This allows us to calculate the Hausdorff dimension of Uβ,N for almost every β > 1. In particular, this improves the main results of Gábor Kallós (1999, 2001). Moreover, we find that the dimension function f(β) = dimHUβ,N fluctuates frequently for β ∈ (1, N).