Large intersection of univoque bases of real numbers

Let $x\in(0,1)$ and $m\in\mathbb N_{\ge 2}$. We consider the set $\Lambda(x)$ of bases $\lambda\in(0, 1/m]$ such that $x=\sum_{i=1}^\infty d_i \lambda^i$ for some (unique) sequence $(d_i)\in\{0,1,\ldots,m-1\}^\mathbb N$. In this paper we show that $\Lambda(x)$ is a topological Cantor set; it has zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that the intersection $\Lambda(x)\cap\Lambda(y)$ has full Hausdorff dimension for any $x, y\in(0,1)$.