The growth rate and dimension theory of beta-expansions

Abstract

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