A two-dimensional univoque set

Martijn de Vrie; Vilmos Komornik

Fundamenta Mathematicae, Tome 215 (2011), p. 175-189 / Harvested from The Polish Digital Mathematics Library

Résumé

Let J ⊂ ℝ² be the set of couples (x,q) with q > 1 such that x has at least one representation of the form $x = \sum_{i = 1}^{\infty} c_{i} q^{- i}$ with integer coefficients $c_{i}$ satisfying $0 \leq c_{i} < q$ , i ≥ 1. In this case we say that $(c_{i}) = c ₁ c ₂ . . .$ 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.

Publié le : 2011-01-01

Zbl 1257.11010

EUDML-ID : urn:eudml:doc:283361

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     year = {2011},
     pages = {175-189},
     zbl = {1257.11010},
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Martijn de Vrie; Vilmos Komornik. A two-dimensional univoque set. Fundamenta Mathematicae, Tome 215 (2011) pp. 175-189. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-fm212-2-4/