Numbers with countable expansions in base of generalized golden ratios

Abstract

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 ${\hat{I}}^2=\mathcal{G}(m)$. 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 $x=\frac{p{\hat{I}}^2+q}{(k+1{)}^n}\in (0,\frac{m}{{\hat{I}}^2-1})$ (where $n,p,q\in \mathbb{Z}$) have ${\aleph }_0$ expansions, while the other elements of $(0,\frac{m}{{\hat{I}}^2-1})$ 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.