2006/09/25 by Martijn de Vries, de Vries, Martijn, Vilmos Komornik +1 · 4 citations
Computer Science · Mathematics · #11A63 #11B83 #37B10 #Combinatorics (math.CO) #Computability, Logic, AI Algorithms #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.CO #math.NT #msc:11A63 #msc:11B83 #msc:37B10 #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.math/0609708
29 pages, some new results added, final version, to appear in Advances in Mathematics
openalex publication_date 2006/09/25 · arxiv created 2008/12/18 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 ∑i=1^∞ ciq-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 Uq of real numbers x having a unique representation of the form ∑i=1^∞ ciq-i=x 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 Uq' consisting of all sequences (ci) of integers ci ∈ [0,q) such that ∑i=1∞ ci q-i ∈ Uq. We determine the numbers r >1 for which the map q ↦ Uq' (defined on (1, ∞)) is constant in a neighborhood of r and the numbers q >1 for which Uq' is a subshift or a subshift of finite type.