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arXiv · 1407.3182

Thue-Morse constant is not badly approximable

Dzmitry Badziahin·Evgeniy Zorin
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Abstract

We prove that Thue-Morse constant $\tau_{TM}=0.01101001..._2$ is not a badly approximable number. Moreover, we prove that $\tau_{TM}(a)=0.01101001..._a$ is not badly approximable for every integer base $a\geq 2$ such that $a$ is not divisible by 15. At the same time we provide a precise formula for convergents of the Laurent series $\tilde{f}_{TM}(z) = z^{-1}\prod_{n=1}^\infty (1-z^{-2^n})$, thus developing further the research initiated by Alf van der Poorten and others.

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Dzmitry Badziahin, Evgeniy Zorin. 2014-07-11. Thue-Morse constant is not badly approximable. https://arxiv.org/abs/1407.3182

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