Thue-Morse constant is not badly approximable

TL;DR

This paper demonstrates that the Thue-Morse constant and its base-$a$ variants (for certain bases) are not badly approximable, providing explicit formulas for their convergents and extending previous research on their properties.

Contribution

It proves the non-badly approximable nature of Thue-Morse constants across multiple bases and derives explicit formulas for their convergents, advancing understanding of their number-theoretic properties.

Findings

Thue-Morse constant is not badly approximable.

Thue-Morse constant in bases not divisible by 15 is not badly approximable.

Provides explicit formulas for convergents of related Laurent series.

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.