2010/03/28 by Martijn de Vries, de Vries, Martijn, Vilmos Komornik +1
Computer Science · Mathematics · #11A63 #11B83 #Advanced Topology and Set Theory #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1003.5335
openalex publication_date 2010/03/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let J ⊂ ℝ2 be the set of couples (x,q) with q>1 such that x has at least one representation of the form x=∑i=1∞ ci q-i with integer coefficients ci satisfying 0 ≤ ci < q, i ≥ 1. In this case we say that (ci)=c1c2... is an expansion of x in base q. Let U be the set of couples (x,q) ∈ J such that x has exactly one expansion in base q. In this paper we deduce some topological and combinatorial properties of the set U. We characterize the closure of U, and we determine its Hausdorff dimension. For (x,q) ∈ J, we also prove new properties of the lexicographically largest expansion of x in base q.