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 $\{0,1,\mathrm{.}\mathrm{.}\mathrm{.},M\}$. We denote by $U_q$ 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 $q\in (1,\infty)$\mathrm{.}Furthermore,weprovethatthedimensionfunction$D:(1,\infty)\to[0,1]$ is continuous, and has a bounded variation. Moreover, it has a Devil’s staircase behavior in $(q^{\mathrm{\prime }},\mathrm{\infty })$, where $q^{\mathrm{\prime }}$ denotes the Komornik–Loreti constant: although $D(q)>D(q^{\mathrm{\prime }})$ for all $q>q^{\mathrm{\prime }}$, we have $D^{\mathrm{\prime }}<0$ a.e. in $(q^{\mathrm{\prime }},\mathrm{\infty })$. During the proofs we improve and generalize a theorem of Erd\H{o}s 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 of bases in which $x=1$ has a unique expansion.