On small bases for which $1$ has countably many expansions

Yuru Zou, Lijin Wang, Jian Lu +1

Let $q\in (1,2)$. A $q$-expansion of a number $x$ in $[0,\frac{1}{q-1}]$ is a sequence $({\delta }_i{)}_{i=1}^{\infty }\in \{0,1{\}}^{\mathbb{N}}$ satisfying $$ x=\sum_{i=1}^\infty\frac{δ_i}{q^i}.$…