On univoque and strongly univoque sets

TL;DR

This paper investigates the properties of univoque and strongly univoque sets in noninteger base expansions, characterizing when the difference set is nonempty, uncountable, or has positive Hausdorff dimension, with implications for Bernoulli convolutions.

Contribution

It introduces the concept of strongly univoque sets, analyzes the set of numbers that are univoque but not strongly univoque, and provides conditions for their size and structure.

Findings

$D_\beta$ is nonempty iff 1 has a unique nonterminating expansion in base $\beta$

$D_\beta$ is uncountable when nonempty

Conditions for $D_\beta$ to have positive Hausdorff dimension

Abstract

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.