On the topological structure of univoque sets
- 关键词：Golden number ; Non-integer bases ; Cantor sets ; Thue– ; Morse sequence ; Beta-expansion
- 刊名：Journal of Number Theory
- 出版年：2007
- 期刊代码：78_0022314x
- 类别：mt
- 出版时间：January 2007
- 卷：122
- 期：1
- 页码：157-183
- 文件大小：219 K
摘要
Erdős, Horváth and Joó discovered some years ago that for some real numbers ffeca" title="Click to view the MathML source">1<q<2 there exists only one sequence ci of zeroes and ones such that ∑ciq−i=1. Subsequently, the set of these numbers was characterized algebraically in [P. Erdős, I. Joó, V. Komornik, Characterization of the unique expansions 1=∑q−ni and related problems, Bull. Soc. Math. France 118 (1990) 377–390] and [V. Komornik, P. Loreti, Subexpansions, superexpansions and uniqueness properties in non-integer bases, Period. Math. Hungar. 44 (2) (2002) 195–216]. We establish an analogous characterization of the closure of . This allows us to clarify the topological structure of these sets: is a countable dense set of , so the latter set is perfect. Moreover, since is known to have zero Lebesgue measure, is a Cantor set.