On long arithmetic progressions in binary Morse-like words

TL;DR

This paper investigates the presence of long arithmetic progressions within the Thue-Morse sequence and related words, using combinatorial methods inspired by van der Waerden's theorem.

Contribution

It extends the understanding of arithmetic progressions in binary Morse-like words, providing new results on their existence and structure.

Findings

Existence of long arithmetic progressions in Thue-Morse words

Generalization to a class of Morse-like words

Method inspired by van der Waerden's combinatorial proof

Abstract

We present results on the existence of long arithmetic progressions in the Thue-Morse word and in a class of generalised Thue-Morse words. Our arguments are inspired by van der Waerden's proof for the existence of arbitrary long monochromatic arithmetic progressions in any finite colouring of the (positive) integers.