Numbers with countable expansions in base of generalized golden ratios

Yuehua Ge, Bo Tan

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 $n\in \mathbb{Z}$…