On two conjectures of Shallit about Thue-Morse-like sequences
Abstract
We study a class of infinite words xk , where k is a positive integer, recently introduced by J. Shallit. This class includes the Thue-Morse sequence x1, the Fibonacci-Thue-Morse sequence x2, and the Allouche-Johnson sequence x3. Shallit stated and for k = 3 proved two conjectures on properties of xk. The first conjecture concerns the factor complexity, the second one the critical exponent of these words. We confirm the validity of both conjectures for every k.
0
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.