Fix 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.