Exceptional digit frequencies and expansions in non-integer bases

Abstract

In this paper we study the set of digit frequencies that are realised by elements of the set of $β$-expansions. The main result of this paper demonstrates that as $β$ approaches $1,$ the set of digit frequencies that occur amongst the set of $β$-expansions fills out the simplex. As an application of our main result, we obtain upper bounds for the local dimension of certain biased Bernoulli convolutions.