Relative bifurcation sets and the local dimension of univoque bases

Fix an alphabet $A=\{0,1,\dots ,M\}$ with $M\in \mathbb{N}$. The univoque set $\mathcal{U}$ 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 investigates how the set $\mathcal{U}$ is distributed over the interval $(1,M+1)$ by determining the limit $$f(q):=\underset{\delta \to 0}{\lim }{\dim }_H(\mathcal{U}\cap (q-\delta ,q+\delta ))$$ for all $q\in (1,M+1)$. We show in particular that $f(q)>0$ if and only if $q\in \bar{\mathcal{U}}\mathrm{\backslash }\mathcal{C}$, where $\mathcal{C}$ 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 $\mathcal{U}$ called {\emph relative bifurcation sets}, and use them to give an explicit expression for the Hausdorff dimension of the intersection of $\mathcal{U}$ with any interval, answering a question of Kalle et al.~[{\emph arXiv:1612.07982; to appear in Acta Arithmetica}, 2018]. Finally, the methods developed in this paper are used to give a complete answer to a question of the first author [{\emph Adv. Math.}, 308:575–598, 2017] about strongly univoque sets.