A two-dimensional univoque set

• Autorzy: Martijn de Vrie , Vilmos Komornik
• Języki publikacji: EN

Abstrakty

EN

Let J ⊂ ℝ² be the set of couples (x,q) with q > 1 such that x has at least one representation of the form $x = ∑_{i=1}^{∞} c_{i} q^{-i}$ with integer coefficients $c_{i}$ satisfying $0 ≤ c_{i} < q$, i ≥ 1. In this case we say that $(c_{i}) = c₁c₂...$ 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.

Kategorie tematyczne

• 11A63: Radix representation; digital problems
• 11B83: Special sequences and polynomials

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Fundamenta Mathematicae
• Rocznik: 2011
• Tom: 212
• Numer: 2
• Strony: 175-189

Daty

wydano

2011

Twórcy

autor

Martijn de Vrie

• Delft University of Technology, Mekelweg 4, 2628 CD Delft, the Netherlands

autor

Vilmos Komornik

• Département de Mathématique, Université de Strasbourg, 7 rue René Descartes, 67084 Strasbourg Cedex, France

Identyfikatory

DOI

10.4064/fm212-2-4