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Exceptional digit frequencies and expansions in non-integer bases (1711.10397v1)
Published 28 Nov 2017 in math.DS and math.NT
Abstract: In this paper we study the set of digit frequencies that are realised by elements of the set of $\beta$-expansions. The main result of this paper demonstrates that as $\beta$ approaches $1,$ the set of digit frequencies that occur amongst the set of $\beta$-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.