Thue-Morse constant is not badly approximable

Badziahin, D.; Zorin, E.

doi:10.1093/imrn/rnu238

Thue-Morse constant is not badly approximable

Badziahin, D.; Zorin, E.

Authors

D. Badziahin

E. Zorin

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

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This is a pre-copyedited, author-produced PDF of an article accepted for publication in International Mathematics Research Notices following peer review. The definitive publisher-authenticated version Badziahin, D. and Zorin, E. (2015) 'Thue-Morse constant is not badly approximable.', International mathematics research notices, 2015(19): 9618-9637 is available online at: http://dx.doi.org/10.1093/imrn/rnu238.