  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.
