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On Abelian repetition threshold

Published online by Cambridge University Press:  14 November 2011

Alexey V. Samsonov  and
Arseny M. Shur
Alexey V. Samsonov
Affiliation:
Institute of Mathematics and Computer Science, Ural Federal University, 620083 pr. Lenina, 51 Ekaterinburg, Russia. vonosmas@gmail.com; Arseny.Shur@usu.ru
Arseny M. Shur
Affiliation:
Institute of Mathematics and Computer Science, Ural Federal University, 620083 pr. Lenina, 51 Ekaterinburg, Russia. vonosmas@gmail.com; Arseny.Shur@usu.ru
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Abstract

We study the avoidance of Abelian powers of words and consider three reasonable generalizations of the notion of Abelian power to fractional powers. Our main goal is to find an Abelian analogue of the repetition threshold, i.e., a numerical value separating k-avoidable and k-unavoidable Abelian powers for each size k of the alphabet. We prove lower bounds for the Abelian repetition threshold for large alphabets and all definitions of Abelian fractional power. We develop a method estimating the exponential growth rate of Abelian-power-free languages. Using this method, we get non-trivial lower bounds for Abelian repetition threshold for small alphabets. We suggest that some of the obtained bounds are the exact values of Abelian repetition threshold. In addition, we provide upper bounds for the growth rates of some particular Abelian-power-free languages.

Keywords

Repetition thresholdformal languagesavoidable repetitionsAbelian powers
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 46 , Issue 1: Special issue dedicated to the 13th "Journées Montoises d'Informatique Théorique" , January 2012 , pp. 147 - 163
Copyright
© EDP Sciences 2011

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