Unique expansions of real numbers
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- 作者：Martijn de Vries ; Vilmos Komornik
- 关键词：Greedy expansion ; Beta-expansion ; Univoque sequence ; Univoque number ; Cantor set ; Thue– ; Morse sequence ; Stable base ; Subshift ; Subshift of finite type
- 刊名：Advances in Mathematics
- 出版年：2009
- 出版时间：1 June 2009
- 年：2009
- 卷：221
- 期：2
- 页码：390-427
- 全文大小：297 K
文摘
It was discovered some years ago that there exist non-integer real numbers q>1 for which only one sequence (ci) of integers ci[0,q) satisfies the equality . The set of such “univoque numbers” has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation.
In this paper we consider for each fixed q>1 the set of real numbers x having a unique representation of the form with integers ci belonging to [0,q). We carry out a detailed topological study of these sets. For instance, we characterize their closures, and we determine those bases q for which is closed or even a Cantor set. We also study the set consisting of all sequences (ci) of integers ci[0,q) such that . We determine the numbers r>1 for which the map (defined on (1,∞)) is constant in a neighborhood of r and the numbers q>1 for which is a subshift or a subshift of finite type.