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 $U_q$ 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 $q\in (1,\infty )$. Furthermore, we prove that the dimension function $D:(1,\infty )\to [0,1]$ is continuous, and has a bounded variation. Moreover, it has a Devil's staircase behavior in $(q^{\prime },\infty )$, where $q^{\prime }$ denotes the Komornik--Loreti constant: although $D(q)>D(q^{\prime })$ for all $q>q^{\prime }$, we have $D^{\prime }<0$ a.e. in $(q^{\prime },\infty )$. During the proofs we improve and generalize a theorem of Erdős 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.

30 pages

References in corpus (1)

• Greedy and quasi-greedy expansions in non-integer bases