Thue, Combinatorics on words, and conjectures inspired by the Thue-Morse sequence

TL;DR

This paper reviews recent findings on the Thue-Morse sequence, discusses open questions and conjectures, and proves a conjecture by Shevelev, advancing understanding in combinatorics on words.

Contribution

It presents new results on the Thue-Morse sequence and resolves a conjecture by Shevelev, contributing to the theoretical understanding of combinatorics on words.

Findings

Proved Shevelev's conjecture related to Thue-Morse sequence

Summarized recent developments in Thue-Morse sequence research

Outlined open questions and future research directions

Abstract

Nous d\'ecrivons quelques r\'esultats r\'ecents sur la suite de Thue-Morse, ainsi que des questions ou conjectures, dont l'une, due \`a Shevelev, est r\'esolue dans cet article. We describe some recent results on the Thue-Morse sequence. We also list open questions and conjectures, one of which is due to Shevelev and proved in this paper.