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Fix an alphabet $A=\{0,1,\dots,M\}$ with $M\in\mathbb{N}$. The univoque set$\mathscr{U}$ of bases $q\in(1,M+1)$ in which the number $1$ has a uniqueexpansion over the alphabet $A$ has been well studied. It has Lebesgue measurezero but Hausdorff dimension one. This paper investigates how the set$\mathscr{U}$ is distributed over the interval $(1,M+1)$ by determining thelimit $$f(q):=\lim_{\delta\to0}\dim_H\big(\mathscr{U}\cap(q-\delta,q+\delta)\big)$$ 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 anuncountable set of Hausdorff dimension zero, and $f$ is continuous at those(and only those) points where it vanishes. Furthermore, we introduce acountable family of pairwise disjoint subsets of $\mathscr{U}$ called {\emphrelative bifurcation sets}, and use them to give an explicit expression for theHausdorff dimension of the intersection of $\mathscr{U}$ with any interval,answering a question of Kalle et al.~[{\emph arXiv:1612.07982; to appear inActa Arithmetica}, 2018]. Finally, the methods developed in this paper are usedto give a complete answer to a question of the first author [{\emph Adv.Math.}, 308:575--598, 2017] about strongly univoque sets.

Relative bifurcation sets and the local dimension of univoque bases