A two-dimensional univoque set

Martijn de Vrie, and Vilmos Komornik. "A two-dimensional univoque set." Fundamenta Mathematicae 212.2 (2011): 175-189. <http://eudml.org/doc/283361>.

@article{MartijndeVrie2011,
abstract = {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.},
author = {Martijn de Vrie, Vilmos Komornik},
journal = {Fundamenta Mathematicae},
keywords = {greedy expansion; beta-expansion; univoque sequence; univoque set; Cantor set; Hausdorff dimension},
language = {eng},
number = {2},
pages = {175-189},
title = {A two-dimensional univoque set},
url = {http://eudml.org/doc/283361},
volume = {212},
year = {2011},
}

TY - JOUR
AU - Martijn de Vrie
AU - Vilmos Komornik
TI - A two-dimensional univoque set
JO - Fundamenta Mathematicae
PY - 2011
VL - 212
IS - 2
SP - 175
EP - 189
AB - 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.
LA - eng
KW - greedy expansion; beta-expansion; univoque sequence; univoque set; Cantor set; Hausdorff dimension
UR - http://eudml.org/doc/283361
ER -