On abelian repetition threshold
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 46 (2012) no. 1, pp. 147-163.

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.

DOI : 10.1051/ita/2011127
Classification : 68Q70, 68R15
Mots-clés : repetition threshold, formal languages, avoidable repetitions, abelian powers 
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Samsonov, Alexey V.; Shur, Arseny M. On abelian repetition threshold. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 46 (2012) no. 1, pp. 147-163. doi : 10.1051/ita/2011127. http://www.numdam.org/articles/10.1051/ita/2011127/

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