Thue-Morse constant is not badly approximable

Badziahin, D.; Zorin, E.

Abstract

We prove that Thue–Morse constant τTM=0.01101001…2 is not a badly approximable number. Moreover, we prove that τTM(a)=0.01101001…a is not badly approximable for every integer base a≥2 such that a is not divisible by 15. At the same time, we provide a precise formula for convergents of the Laurent series f~TM(z)=z−1∏∞n=1(1−z−2n), thus developing further the research initiated by Alf van der Poorten and others.

Citation

Badziahin, D., & Zorin, E. (2015). Thue-Morse constant is not badly approximable. International Mathematics Research Notices, 2015(19), 9618-9637. https://doi.org/10.1093/imrn/rnu238

• Journal Article Type: Article
• Acceptance Date: Nov 5, 2014
• Online Publication Date: Dec 4, 2014
• Publication Date: Nov 1, 2015
• Deposit Date: Nov 26, 2014
• Publicly Available Date: Dec 18, 2014
• Journal: International Mathematics Research Notices
• Print ISSN: 1073-7928
• Electronic ISSN: 1687-0247
• Publisher: Oxford University Press
• Peer Reviewed: Peer Reviewed
• Volume: 2015
• Issue: 19
• Pages: 9618-9637
• DOI: https://doi.org/10.1093/imrn/rnu238
• Public URL: https://durham-repository.worktribe.com/output/1449749