The Thue-Morse Transform

TL;DR

This paper introduces the Thue--Morse transform based on evil and odious numbers, revealing its properties, solutions to the Prouhet--Tarry--Escott problem, and extensions to d-ary and Fibonacci-based partitions.

Contribution

It defines the Thue--Morse transform, explores its properties, and extends classical partitions to new frameworks including d-ary and Fibonacci analogues.

Findings

Iterated sequences form a family with a dyadic structure.

Solutions to the Prouhet--Tarry--Escott problem are extended.

Exact factor complexity is determined for Mersenne levels.

Abstract

We introduce the Thue--Morse transform, a transform on binary sequences defined through their evil and odious numbers, namely the positions of $0$'s and $1$'s, respectively, and prove that its iterates on the classical Thue--Morse sequence form an explicit family of binary sequences with a clear dyadic structure, extending the classical Prouhet--Thue--Morse partition. We show that these iterated sequences yield broad new families of solutions to the Prouhet--Tarry--Escott problem, extending Prouhet's classical digit-sum construction rather than producing ideal solutions. We prove functional equations for the associated generalized evil and odious numbers that extend the classical composition formulas for evil and odious numbers. For Mersenne levels we determine the factor complexity completely, proving an exact hierarchical piecewise formula via a desubstitution argument. We also formulate two extensions beyond the basic dyadic setting: a $d$-ary version of the same mechanism, and a Fibonacci analogue of the Prouhet partition based on Zeckendorf numeration.