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

📝 Original Info

• Title: Monochromatic arithmetic progressions in the Fibonacci, Thue-Morse, and Rudin-Shapiro words
• ArXiv ID: 2501.05830
• Date: 2025-01-10
• Authors: Gandhar Joshi, Dan Rust

📝 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.

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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.

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