2006/11/19 by Geneviève Paquin, Paquin, G., Laurent Vuillon +1
Computer Science · Materials Science · Mathematics · #11B50 #68R05 #68R15 #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #O5A17 #Quasicrystal Structures and Properties #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.math/0611576
openalex publication_date 2006/11/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
It is well known that Sturmian sequences are the aperiodic sequences that are balanced over a 2-letter alphabet. They are also characterized by their complexity: they have exactly (n+1) factors of length n. One possible generalization of Sturmian sequences is the set of infinite sequences over a k-letter alphabet, k ≥ 3, which are closed under reversal and have at most one right special factor for each length. This is the set of episturmian sequences. These are not necessarily balanced over a k-letter alphabet, nor are they necessarily aperiodic. In this paper, we characterize balanced episturmian sequences, periodic or not, and prove Fraenkel's conjecture for the class of episturmian sequences. This conjecture was first introduced in number theory and has remained unsolved for more than 30 years. It states that for a fixed k> 2, there is only one way to cover \Z by k Beatty sequences. The problem can be translated to combinatorics on words: for a k-letter alphabet, there exists only one balanced sequence up to letter permutation that has different letter frequencies.