Unique expansions of real numbers

Abstract

It was discovered some years ago that there exist non-integer real numbers $q>1$ for which only one sequence $(c_i)$ of integers $c_i\in [0,q)$ satisfies the equality ${\sum }_{i=1}^{\infty }{c}_i{q}^{-i}=1$. 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 ${\mathcal{U}}_q$ of real numbers $x$ having a unique representation of the form ${\sum }_{i=1}^{\infty }{c}_i{q}^{-i}=x$ with integers $c_i$ 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 ${\mathcal{U}}_q$ is closed or even a Cantor set. We also study the set ${\mathcal{U}}_q^{\prime }$ consisting of all sequences $(c_i)$ of integers $c_i\in [0,q)$ such that ${\sum }_{i=1}^{\infty }{c}_i{q}^{-i}\in {\mathcal{U}}_q$. We determine the numbers $r>1$ for which the map $q\mapsto {\mathcal{U}}_q^{\prime }$ (defined on $(1,\infty )$) is constant in a neighborhood of $r$ and the numbers $q>1$ for which ${\mathcal{U}}_q^{\prime }$ is a subshift or a subshift of finite type.

29 pages, some new results added, final version, to appear in Advances in Mathematics