On univoque and strongly univoque sets

Much has been written about expansions of real numbers in noninteger bases. Particularly, for a finite alphabet $\{0,1,\dots,\alpha\}$ and a real number (base) $1<\beta<\alpha+1$, the so-called {\em univoque set} of numbers which have a unique expansion in base $\beta$ has garnered a great deal of attention in recent years. Motivated by recent applications of $\beta$-expansions to Bernoulli convolutions and a certain class of self-affine functions, we introduce the notion of a {\em strongly univoque} set. We study in detail the set $D_\beta$ of numbers which are univoque but not strongly univoque. Our main result is that $D_\beta$ is nonempty if and only if the number $1$ has a unique nonterminating expansion in base $\beta$, and in that case, $D_\beta$ is uncountable. We give a sufficient condition for $D_\beta$ to have positive Hausdorff dimension, and show that, on the other hand, there are infinitely many values of $\beta$ for which $D_\beta$ is uncountable but of Hausdorff dimension zero.