On univoque and strongly univoque sets

Abstract

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