On generalized Thue-Morse functions and their values

Badziahin, D.; Zorin, E.

Abstract

In this paper we extend and generalize, up to a natural bound of the method, our previous work Badziahin and Zorin [‘Thue–Morse constant is not badly approximable’, Int. Math. Res. Not. IMRN 19 (2015), 9618–9637] where we proved, among other things, that the Thue–Morse constant is not badly approximable. Here we consider Laurent series defined with infinite products fd(x) = Q∞ n=0 (1 − x −d n ), d ∈ N, d ≥ 2, which generalize the generating function f2(x) of the Thue–Morse number, and study their continued fraction expansion. In particular, we show that the convergents of x −d+1 fd(x) have a regular structure. We also address the question of whether the corresponding Mahler numbers fd(a) ∈ R, a, d ∈ N, a, d ≥ 2, are badly approximable.

Citation

Badziahin, D., & Zorin, E. (2020). On generalized Thue-Morse functions and their values. Journal of the Australian Mathematical Society, 108(2), 177-201. https://doi.org/10.1017/s1446788718000459

• Journal Article Type: Article
• Acceptance Date: Sep 12, 2018
• Online Publication Date: Mar 11, 2019
• Publication Date: Apr 30, 2020
• Deposit Date: Nov 15, 2019
• Publicly Available Date: Nov 15, 2019
• Journal: Journal of the Australian Mathematical Society
• Print ISSN: 1446-7887
• Electronic ISSN: 1446-8107
• Publisher: Cambridge University Press
• Peer Reviewed: Peer Reviewed
• Volume: 108
• Issue: 2
• Pages: 177-201
• DOI: https://doi.org/10.1017/s1446788718000459
• Public URL: https://durham-repository.worktribe.com/output/1283511
• Related Public URLs: http://eprints.whiterose.ac.uk/135616/