A4 Refereed article in a conference publication
Every nonnegative real number is an abelian critical exponent
Authors: Peltomäki Jarkko, Whiteland Markus A.
Editors: Mercaş Robert, Reidenbach Daniel
Conference name: International Conference on Combinatorics on Words
Publication year: 2019
Journal: Lecture Notes in Computer Science
Book title : Combinatorics on Words: 12th International Conference, WORDS 2019, Loughborough, UK, September 9–13, 2019, Proceedings
Series title: Lecture Notes in Computer Science
Volume: 11682
First page : 275
Last page: 285
ISBN: 978-3-030-28795-5
eISBN: 978-3-030-28796-2
DOI: https://doi.org/10.1007/978-3-030-28796-2_22
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/41978810
The abelian critical exponent of an infinite word $w$ is defined as the maximum ratio between the exponent and the period of an abelian power occurring in $w$. It was shown by Fici et al. that the set of finite abelian critical exponents of Sturmian words coincides with the Lagrange spectrum. This spectrum contains every large enough positive real number. We construct words whose abelian critical exponents fill the remaining gaps, that is, we prove that for each nonnegative real number $\theta$ there exists an infinite word having abelian critical exponent $\theta$. We also extend this result to the $k$-abelian setting.
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