Monochromatic arithmetic progressions in the Fibonacci, Thue-Morse, and Rudin-Shapiro words

TL;DR

This paper studies the longest monochromatic arithmetic progressions in specific infinite words, providing formulas and classifications using dynamical systems and computer-assisted proofs, extending known results for Fibonacci, Thue-Morse, and Rudin-Shapiro words.

Contribution

It offers a complete classification of progression lengths in the Fibonacci word and extends results to Thue-Morse and Rudin-Shapiro words using novel dynamical and computational methods.

Findings

Complete classification of progression lengths in Fibonacci word

Extended results for Thue-Morse and Rudin-Shapiro words

Combined dynamical systems and computer-assisted approaches

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.