On generalized Thue-Morse functions and their values

Abstract

This paper naturally extends and generalizes our previous work "Thue-Morse constant is not badly approximable", arXiv:1407.3182 [math.NT]. Here we consider the Laurent series $f_d(x)={\prod }_{n=0}^{\infty }(1-x^{-d^n})$, $d\in \mathbb{N}$, $d\ge 2$ which generalize the generating function $f_2(x)$ of the Thue-Morse number, and study their continued fraction expansion. In particular, we show that the convergents of $x^{-d+1}{f}_d(x)$ have quite a regular structure. We address as well the question whether the corresponding Mahler numbers $f_d(a)\in \mathbb{R}$, $a,d\in \mathbb{N}$, $a,d\ge 2$, are badly approximable.

arXiv comment20 pages