Hausdorff dimension of univoque sets and Devil's staircase

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 U, the set of real numbers having a unique expansion. Completing many former investigations, we give a formula for the Hausdorff dimension D(q) of u(q) for each q epsilon (1, infinity). Furthermore, we prove that the dimension function D : (1, infinity) [0,1] is continuous, and has bounded variation. Moreover, it has a Devil's staircase behavior in (q', infinity), where q' denotes the Komornik-Loreti constant: although D(q) > D(q') for all q > q', we have D' < 0 a.e. in (qi, infinity). During the proofs we improve and generalize a theorem of Erdds et al. on the existence of large blocks of zeros in beta-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. (C) 2016 Elsevier Inc. All rights reserved.