Unique expansions of real numbers |
| |
Authors: | Martijn de Vries Vilmos Komornik |
| |
Institution: | a Delft University of Technology, Mekelweg 4, 2628 CD Delft, The Netherlands b Département de Mathématique, Université Louis Pasteur, 7 rue René Descartes, 67084 Strasbourg Cedex, France |
| |
Abstract: | 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 Uq 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 Uq 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. |
| |
Keywords: | primary 11A63 secondary 11B83 37B10 |
本文献已被 ScienceDirect 等数据库收录! |
|