Hausdorff dimension of univoque sets and Devil's staircase

Abstract

We fix a positive integer M, and we consider expansions in arbitrary real bases q>1 over the alphabet {0,1,…,M}. We denote by Uq the set of real numbers having a unique expansion. Completing many former investigations, we give a formula for the Hausdorff dimension D(q) of Uq for each q∈(1,∞). Furthermore, we prove that the dimension function D:(1,∞)→[0,1] is continuous, and has bounded variation. Moreover, it has a Devil's staircase behavior in (q′,∞), where q′ denotes the Komornik–Loreti constant: although D(q)>D(q′) for all q>q′, we have D′<0 a.e. in (q′,∞). During the proofs we improve and generalize a theorem of Erdős et al. on the existence of large blocks of zeros in β-expansions, and we determine for all M the Lebesgue measure and the Hausdorff dimension of the set U of bases in which x=1 has a unique expansion.