1. Univoque bases of real numbers: Simply normal bases, irregular bases and multiple rationals.
- Author
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Hu, Yu, Huang, Yan, and Kong, Derong
- Subjects
- *
FRACTAL dimensions , *REAL numbers , *CANTOR sets - Abstract
Given a positive integer M and a real number x ∈ (0 , 1 ] , we call q ∈ (1 , M + 1 ] a univoque simply normal base of x if there exists a unique simply normal sequence (d i) ∈ { 0 , 1 , ... , M } N such that x = ∑ i = 1 ∞ d i q − i. Similarly, a base q ∈ (1 , M + 1 ] is called a univoque irregular base of x if there exists a unique sequence (d i) ∈ { 0 , 1 , ... , M } N such that x = ∑ i = 1 ∞ d i q − i and the sequence (d i) has no digit frequency. Let U S N (x) and U I r (x) be the sets of univoque simply normal bases and univoque irregular bases of x , respectively. In this paper we show that for any x ∈ (0 , 1 ] both U S N (x) and U I r (x) have full Hausdorff dimension. Furthermore, given finitely many rationals x 1 , x 2 , ... , x ℓ ∈ (0 , 1 ] so that each x i has a finite expansion in base M + 1 , we show that there exists a full Hausdorff dimensional set of q ∈ (1 , M + 1 ] such that each x i has a unique expansion in base q. [ABSTRACT FROM AUTHOR]
- Published
- 2023
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