A two-dimensional univoque set

Let $\mathbf{J} \subset \mathbb{R}^2$ 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 $0 \leq c_i < q$, $i \ge 1$. In this case we say that $(c_i)=c_1c_2...$ is an expansion of $x$ in base $q$. Let $\mathbf{U}$ be the set of couples $(x,q) \in \mathbf{J}$ such that $x$ has exactly one expansion in base $q$. In this paper we deduce some topological and combinatorial properties of the set $\mathbf{U}$. We characterize the closure of $\mathbf{U}$, and we determine its Hausdorff dimension. For $(x,q) \in \mathbf{J}$, we also prove new properties of the lexicographically largest expansion of $x$ in base $q$.