  In a recent paper of Feng and Sidorov they show that for
$\beta\in(1,\frac{1+\sqrt{5}}{2})$ the set of $\beta$-expansions grows
exponentially for every $x\in(0,\frac{1}{\beta-1})$. In this paper we study
this growth rate further. We also consider the set of $\beta$-expansions from a
dimension theory perspective.
