Monochromatic arithmetic progressions in the Fibonacci, Thue-Morse, and Rudin-Shapiro words
Abstract
We investigate the lengths and starting positions of the longest monochromatic arithmetic progressions for a fixed difference in the Fibonacci word. We provide a complete classification for their lengths in terms of a simple formula. Our strongest results are proved using methods from dynamical systems, especially the dynamics of circle rotations. We also employ computer-based methods in the form of the automatic theorem-proving software Walnut. This allows us to extend recent results concerning similar questions for the Thue-Morse word and the Rudin-Shapiro word. This also allows us to obtain some results for the Fibonacci word that do not seem to be amenable to dynamical methods.
Keywords
Cite
@article{arxiv.2501.05830,
title = {Monochromatic arithmetic progressions in the Fibonacci, Thue-Morse, and Rudin-Shapiro words},
author = {Gandhar Joshi and Dan Rust},
journal= {arXiv preprint arXiv:2501.05830},
year = {2025}
}
Comments
Changes made to address feedback from anonymous referee. Accepted for publication in Theoretical Computer Science