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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
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 | ||||||||
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| 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: |
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| 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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