Relative bifurcation sets and the local dimension of univoque bases

Abstract

AbstractFix an alphabet $A=\{0,1,\ldots ,M\}$ with $M\in \mathbb{N}$. The univoque set $\mathscr{U}$ of bases $q\in (1,M+1)$ in which the number $1$ has a unique expansion over the alphabet $A$ has been well studied. It has Lebesgue measure zero but Hausdorff dimension one. This paper describes how the points in the set $\mathscr{U}$ are distributed over the interval $(1,M+1)$ by determining the limit $$\begin{eqnarray}f(q):=\lim _{\unicode[STIX]{x1D6FF}\rightarrow 0}\dim _{\text{H}}(\mathscr{U}\cap (q-\unicode[STIX]{x1D6FF},q+\unicode[STIX]{x1D6FF}))\end{eqnarray}$$ for all $q\in (1,M+1)$. We show in particular that $f(q)>0$ if and only if $q\in \overline{\mathscr{U}}\backslash \mathscr{C}$, where $\mathscr{C}$ is an uncountable set of Hausdorff dimension zero, and $f$ is continuous at those (and only those) points where it vanishes. Furthermore, we introduce a countable family of pairwise disjoint subsets of $\mathscr{U}$ called relative bifurcation sets, and use them to give an explicit expression for the Hausdorff dimension of the intersection of $\mathscr{U}$ with any interval, answering a question of Kalle et al [On the bifurcation set of unique expansions. Acta Arith. 188 (2019), 367–399]. Finally, the methods developed in this paper are used to give a complete answer to a question of the first author [On univoque and strongly univoque sets. Adv. Math.308 (2017), 575–598] on strongly univoque sets.