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 Kalleet 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.