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

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.