2025/01/10 by Joshi, Gandhar, D. M. Rust, Rust, Dan
Mathematics · Physics and Astronomy · #11B85 #37B10 #52C23 #68Q45 #Advanced Mathematical Theories and Applications #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Computer and information sciences #FOS: Mathematics #Formal Languages and Automata Theory (cs.FL) #History and Theory of Mathematics #Mathematics and Applications
paper · pdf · doi:10.48550/arxiv.2501.05830
openalex publication_date 2025/01/10 · openalex created_date 2025/01/14 · openalex updated_date 2026/07/28
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.