  Sidorov and Vershik showed that in base $G=\frac{\sqrt{5}+1}{2}$ and with the
digits $0,1$ the numbers $x=nG ~(\text {mod} 1)$ have $\aleph_{0}$ expansions
for any $n\in\mathbb{Z}$, while the other elements of $(0, \frac{1}{G-1})$ have
$2^{\aleph_{0}}$ expansions. In this paper, we generalize this result to the
generalized golden ratio base $\beta=\mathcal{G}(m)$. With the digit-set
$\{0,1,\cdots, m\}$, if $m=2k+1$,
$\mathcal{G}(m)=\frac{k+1+\sqrt{k^{2}+6k+5}}{2}$, the numbers
$x=\frac{p\beta+q}{(k+1)^{n}}\in(0, \frac{m}{\beta-1})$ (where $n, p,
q\in\mathbb{Z}$) have $\aleph_{0}$ expansions, while the other elements of $(0,
\frac{m}{\beta-1})$ have $2^{\aleph_{0}}$ expansions; if $m=2k$,
$\mathcal{G}(m)=k+1$, the numbers with countably many expansions are
$\frac{p}{(k+1)^{n}}\in(0, 2) ~(n, p\in\mathbb{N}\cup\{0\})$. This solves an
open question by Baker.
