The Thue-Morse Transform
Abstract
We define the Thue-Morse transform T on a class of infinite binary words. It sends the alternating word a0 = 010101... to the Thue-Morse sequence. We then study its orbit am = Tm(a0) as well as the sequences um and vm giving respectively the positions of the ones and the zeros in am. We obtain an explicit formula for am and deduce Prouhet-Tarry-Escott identities for the partitions induced by um and vm. We also give composition formulas for um and vm, and a full description of the factor complexity of am.
Turn this paper into a full lesson
ArcXiv compiles a staged curriculum from this paper: 8-12 lessons across beginner → advanced, synthesised section guides, visuals, flashcards, a quiz, exercises, and on-demand deep dives per section. Grounded in the abstract, never invented.