On generalized Thue-Morse functions and their values

TL;DR

This paper studies generalized Thue-Morse functions defined by infinite products, analyzing their continued fractions and approximation properties, and extends previous work on their number-theoretic characteristics.

Contribution

It introduces a broader class of Thue-Morse related functions, examines their continued fraction structure, and investigates their approximation properties, extending prior results.

Findings

Convergents of the functions have a regular continued fraction structure

The paper addresses whether the associated Mahler numbers are badly approximable

Extends previous work on Thue-Morse constants to a generalized setting

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 $f_d(x) = \prod_{n=0}^\infty (1 - x^{-d^n})$, $d\in\mathbb{N}$, $d\geq 2$ 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 $f_d(a)\in\mathbb{R}$, $a,d\in\mathbb{N}$, $a,d\geq 2$, are badly approximable.