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Numbers with simply normal β-expansions

Published online by Cambridge University Press:  26 April 2018

SIMON BAKER  and
DERONG KONG
SIMON BAKER
Affiliation:
Mathematical Institute, University of Warwick, Coventry, CV4 7ALU.K. e-mail: simonbaker412@gmail.com
DERONG KONG
Affiliation:
Mathematical Institute, University of Leiden, PO Box 9512, 2300 RA Leiden, The Netherlands. e-mail: d.kong@math.leidenuniv.nl
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Abstract

In [6] the first author proved that for any β ∈ (1, βKL) every x ∈ (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 an x ∈ (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 polynomial x3 − 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], every x ∈ (0, 1/(β − 1)) has a simply normal β-expansion. For completion, we provide a proof that for any β ∈ (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.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 167 , Issue 1 , July 2019 , pp. 171 - 192
Copyright
Copyright © Cambridge Philosophical Society 2018 

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