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.
- Expansions in non-integer bases
- Digit frequencies
- Bernoulli convolutions