Consecutive Power Occurrences in Sturmian Words

Jason Bell; Chris Schulz; Jeffrey Shallit

doi:10.5802/crmath.644

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Consecutive Power Occurrences in Sturmian Words
[Occurrences consécutives de puissance dans les mots sturmiens]

Jason Bell ¹ ; Chris Schulz ¹ ; Jeffrey Shallit ²

¹ University of Waterloo, Department of Pure Mathematics, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada
² University of Waterloo, School of Computer Science, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1273-1278.

Résumé
Abstract

Nous montrons que la distance entre deux positions finales consécutives de cubes apparaissant dans un mot sturmien est toujours inférieure ou égale à $10$ et que cette valeur est optimale, étendant ainsi un résultat de Rampersad, qui a démontré que cette distance est majorée par $9$ pour le mot de Fibonacci. Nous donnons ensuite un résultat général montrant que pour tout $e \in [1, (5 + \sqrt{5}) / 2)$ il existe un entier naturel $N$ , dépendant uniquement de $e$ , tel que la distance entre deux positions finales consécutives de puissances $e$ apparaissant dans un mot sturmien est uniformément majorée par $N$ .

We show that every Sturmian word has the property that the distance between consecutive ending positions of cubes occurring in the word is always bounded by $10$ and this bound is optimal, extending a result of Rampersad, who proved that the bound $9$ holds for the Fibonacci word. We then give a general result showing that for every $e \in [1, (5 + \sqrt{5}) / 2)$ there is a natural number $N$ , depending only on $e$ , such that every Sturmian word has the property that the distance between consecutive ending positions of $e$ -powers occurring in the word is uniformly bounded by $N$ .

Reçu le : 2024-02-19
Révisé le : 2024-03-26
Accepté le : 2024-04-21
Publié le : 2024-11-05

DOI : 10.5802/crmath.644

Classification : 68R15
Keywords: Sturmian word, cube, periodicity, balanced word
Mot clés : Mot Sturmien, cube, périodicité, mot équilibré

Affiliations des auteurs :

Jason Bell ¹ ; Chris Schulz ¹ ; Jeffrey Shallit ²

¹ University of Waterloo, Department of Pure Mathematics, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada
² University of Waterloo, School of Computer Science, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Jason Bell and Chris Schulz and Jeffrey Shallit},
     title = {Consecutive {Power} {Occurrences} in {Sturmian} {Words}},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1273--1278},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.644},
     language = {en},
}

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Jason Bell; Chris Schulz; Jeffrey Shallit. Consecutive Power Occurrences in Sturmian Words. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1273-1278. doi : 10.5802/crmath.644. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.644/

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