Thue-Morse constant is not badly approximable

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.