Numbers with simply normal $β$-expansions

Abstract

In [Bak] the first author proved that for any $β\in (1,β_{KL})$ every $x\in(0,\frac{1}{β-1})$ has a simply normal $β$-expansion, where $β_{KL}\approx 1.78723$ is the Komornik-Loreti constant. This result is complemented by an observation made in [JSS], where it was shown that whenever $β\in (β_T, 2]$ there exists an $x\in(0,\frac{1}{β-1})$ with a unique $β$-expansion, and this expansion is not simply normal. Here $β_T\approx 1.80194$ is the unique zero in $(1,2]$ of the polynomial $x^3-x^2-2x+1$. This leaves a gap in our understanding within the interval $[β_{KL}, β_T]$. In this paper we fill this gap and prove that for any $β\in (1,β_T],$ every $x\in(0,\frac{1}{β-1})$ has a simply normal $β$-expansion. For completion, we provide a proof that for any $β\in(1,2)$, Lebesgue almost every $x$ has a simply normal $β$-expansion. We also give examples of $x$ with multiple $β$-expansions, none of which are simply normal. Our proofs rely on ideas from combinatorics on words and dynamical systems.