Consecutive Power Occurrences in Sturmian Words

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 gen...

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Main Authors: Bell, Jason, Schulz, Chris, Shallit, Jeffrey
Format: Article
Language:English
Published: Académie des sciences 2024-11-01
Series:Comptes Rendus. Mathématique
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Online Access:https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.644/
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author Bell, Jason
Schulz, Chris
Shallit, Jeffrey
author_facet Bell, Jason
Schulz, Chris
Shallit, Jeffrey
author_sort Bell, Jason
collection DOAJ
description 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 $N$.
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institution Kabale University
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publishDate 2024-11-01
publisher Académie des sciences
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series Comptes Rendus. Mathématique
spelling doaj-art-24941768581c477892db12aeccec675e2025-02-07T11:23:32ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692024-11-01362G101273127810.5802/crmath.64410.5802/crmath.644Consecutive Power Occurrences in Sturmian WordsBell, Jason0Schulz, Chris1Shallit, Jeffrey2University of Waterloo, Department of Pure Mathematics, 200 University Avenue West, Waterloo, Ontario N2L 3G1, CanadaUniversity of Waterloo, Department of Pure Mathematics, 200 University Avenue West, Waterloo, Ontario N2L 3G1, CanadaUniversity of Waterloo, School of Computer Science, 200 University Avenue West, Waterloo, Ontario N2L 3G1, CanadaWe 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 $N$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.644/Sturmian wordcubeperiodicitybalanced word
spellingShingle Bell, Jason
Schulz, Chris
Shallit, Jeffrey
Consecutive Power Occurrences in Sturmian Words
Comptes Rendus. Mathématique
Sturmian word
cube
periodicity
balanced word
title Consecutive Power Occurrences in Sturmian Words
title_full Consecutive Power Occurrences in Sturmian Words
title_fullStr Consecutive Power Occurrences in Sturmian Words
title_full_unstemmed Consecutive Power Occurrences in Sturmian Words
title_short Consecutive Power Occurrences in Sturmian Words
title_sort consecutive power occurrences in sturmian words
topic Sturmian word
cube
periodicity
balanced word
url https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.644/
work_keys_str_mv AT belljason consecutivepoweroccurrencesinsturmianwords
AT schulzchris consecutivepoweroccurrencesinsturmianwords
AT shallitjeffrey consecutivepoweroccurrencesinsturmianwords