A two-dimensional univoque set
Let J ⊂ ℝ² be the set of couples (x,q) with q > 1 such that x has at least one representation of the form with integer coefficients satisfying , i ≥ 1. In this case we say that 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...