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Numbers with simply normal beta-expansions

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Baker, Simon and Kong, Derong (2019) Numbers with simply normal beta-expansions. Mathematical Proceedings of the Cambridge Philosophical Society, 167 (1). pp. 171-192. doi:10.1017/S0305004118000270 ISSN 0305-0041.

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Official URL: https://doi.org/10.1017/S0305004118000270

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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.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Journal or Publication Title: Mathematical Proceedings of the Cambridge Philosophical Society
Publisher: Cambridge University Press
ISSN: 0305-0041
Official Date: July 2019
Dates:
DateEvent
July 2019Published
26 April 2018Available
26 March 2018Accepted
Volume: 167
Number: 1
Page Range: pp. 171-192
DOI: 10.1017/S0305004118000270
Status: Peer Reviewed
Publication Status: Published
Access rights to Published version: Restricted or Subscription Access
Date of first compliant deposit: 16 April 2018
Date of first compliant Open Access: 26 October 2018
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