Thue-Morse constant is not badly approximable

Abstract

We prove that Thue-Morse constant ${\tau }_{TM}=\mathrm{0.01101001..}{.}_2$ is not a badly approximable number. Moreover, we prove that ${\tau }_{TM}(a)=\mathrm{0.01101001..}{.}_a$ is not badly approximable for every integer base $a\ge 2$ 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.