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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Académie des sciences
2024-11-01
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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 $N$. |
format | Article |
id | doaj-art-24941768581c477892db12aeccec675e |
institution | Kabale University |
issn | 1778-3569 |
language | English |
publishDate | 2024-11-01 |
publisher | Académie des sciences |
record_format | Article |
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 |