On Abelian repetition threshold

Alexey V. Samsonov; Arseny M. Shur

RAIRO - Theoretical Informatics and Applications (2012)

  • Volume: 46, Issue: 1, page 147-163
  • ISSN: 0988-3754

Abstract

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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.

How to cite

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Samsonov, Alexey V., and Shur, Arseny M.. "On Abelian repetition threshold." RAIRO - Theoretical Informatics and Applications 46.1 (2012): 147-163. <http://eudml.org/doc/222002>.

@article{Samsonov2012,
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.},
author = {Samsonov, Alexey V., Shur, Arseny M.},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Repetition threshold; formal languages; avoidable repetitions; Abelian powers; repetition threshold; abelian powers},
language = {eng},
month = {3},
number = {1},
pages = {147-163},
publisher = {EDP Sciences},
title = {On Abelian repetition threshold},
url = {http://eudml.org/doc/222002},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Samsonov, Alexey V.
AU - Shur, Arseny M.
TI - On Abelian repetition threshold
JO - RAIRO - Theoretical Informatics and Applications
DA - 2012/3//
PB - EDP Sciences
VL - 46
IS - 1
SP - 147
EP - 163
AB - 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.
LA - eng
KW - Repetition threshold; formal languages; avoidable repetitions; Abelian powers; repetition threshold; abelian powers
UR - http://eudml.org/doc/222002
ER -

References

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