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.
Categories: math.NT math.DS