Equations of the form $t(x+a)=t(x)$ and $t(x+a)=1-t(x)$ for Thue-Morse sequence

TL;DR

This paper develops a recursive algorithm to find solutions to specific equations involving the Thue-Morse sequence and introduces open problems and conjectures related to these equations.

Contribution

It provides a novel recursive method for constructing solution sets of equations involving the Thue-Morse sequence and formulates open problems and conjectures.

Findings

Recursive algorithm for solutions of $t(x+a)=t(x)$ and $t(x+a)=1-t(x)$.

Open problem posed regarding the structure of solutions.

Two conjectures related to the properties of solutions.

Abstract

For every $a\geq1$ we give a recursion algorithm of building of set of solutions of equations of the form $t(x+a)=t(x)$ and $t(x+a)=1-t(x),$ where $\{t(n)\}$ is Thue-Morse sequence. We pose an open problem and two conjectures.