Multiple expansions of real numbers with digits set $\{0,1,q\}$

Abstract: For $q>1$ we consider expansions in base $q$ over the alphabet ${0,1,q}$. Let $\mathcal{U}q$ be the set of $x$ which have a unique $q$-expansions. For $k=2, 3,\cdots,\aleph_0$ let $\mathcal{B}_k$ be the set of bases $q$ for which there exists $x$ having $k$ different $q$-expansions, and for $q\in \mathcal{B}_k$ let $\mathcal{U}_q^{(k)}$ be the set of all such $x$'s which have $k$ different $q$-expansions. In this paper we show that [ \mathcal{B}{\aleph_0}=[2,\infty),\quad \mathcal{B}k=(q_c,\infty)\quad \textrm{for any}\quad k\ge 2, ] where $q_c\approx 2.32472$ is the appropriate root of $x^3-3x^2+2x-1=0$. Moreover, we show that for any positive integer $k\ge 2$ and any $q\in\mathcal{B}{k}$ the Hausdorff dimensions of $\mathcal{U}_q^{(k)}$ and $\mathcal{U}_q$ are the same, i.e., [ \dim_H\mathcal{U}_q^{(k)}=\dim_H\mathcal{U}_q\quad\textrm{for any}\quad k\ge 2. ] Finally, we conclude that the set of $x$ having a continuum of $q$-expansions has full Hausdorff dimension.