Numbers with simply normal β-expansions

Abstract

AbstractIn [6] the first author proved that for any β ∈ (1, βKL) everyx∈ (0, 1/(β − 1)) has a simply normal β-expansion, where βKL≈ 1.78723 is the Komornik–Loreti constant. This result is complemented by an observation made in [22], where it was shown that whenever β ∈ (βT, 2] there exists anx∈ (0, 1/(β − 1)) with a unique β-expansion, and this expansion is not simply normal. Here βT≈ 1.80194 is the unique zero in (1, 2] of the polynomialx3−x2− 2x+ 1. This leaves a gap in our understanding within the interval [βKL, βT]. In this paper we fill this gap and prove that for any β ∈ (1, βT], everyx∈ (0, 1/(β − 1)) has a simply normal β-expansion. For completion, we provide a proof that for any β ∈ (1, 2), Lebesgue almost everyxhas a simply normal β-expansion. We also give examples ofxwith multiple β-expansions, none of which are simply normal.Our proofs rely on ideas from combinatorics on words and dynamical systems.