On abelian repetition threshold
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 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
Keywords: 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, Volume 46 (2012) no. 1, pp. 147-163. doi : 10.1051/ita/2011127. http://www.numdam.org/articles/10.1051/ita/2011127/

[1] A. Aberkane, J.D. Currie and N. Rampersad, The number of ternary words avoiding Abelian cubes grows exponentially. J. Integer Seq. 7 (2004) 13 (electronic only). | MR | Zbl

[2] F.-J. Brandenburg, Uniformly growing k-th power free homomorphisms. Theoret. Comput. Sci. 23 (1983) 69-82. | MR | Zbl

[3] A. Carpi, On the number of Abelian square-free words on four letters. Discrete Appl. Math. 81 (1998) 155-167. | MR | Zbl

[4] A. Carpi, On Dejean's conjecture over large alphabets. Theoret. Comput. Sci. 385 (2007) 137-151. | MR | Zbl

[5] M. Crochemore, F. Mignosi and A. Restivo, Automata and forbidden words. Inf. Process. Lett. 67 (1998) 111-117. | MR

[6] J.D. Currie, The number of binary words avoiding Abelian fourth powers grows exponentially. Theoret. Comput. Sci. 319 (2004) 441-446. | MR | Zbl

[7] J.D. Currie and N. Rampersad, A proof of Dejean's conjecture. Math. Comput. 80 (2011) 1063-1070. | Zbl

[8] F. Dejean, Sur un théorème de Thue. J. Comb. Th. (A) 13 (1972) 90-99. | MR | Zbl

[9] F.M. Dekking, Strongly non-repetitive sequences and progression-free sets. J. Comb. Th. (A) 27 (1979) 181-185. | MR | Zbl

[10] P. Erdös, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. 6 (1961) 221-264. | MR | Zbl

[11] V. Keränen, Abelian squares are avoidable on 4 letters, in Proc. ICALP'92. Lect. Notes Comput. Sci. 623 (1992) 41-52. | MR

[12] V. Keränen, A powerful abelian square-free substitution over 4 letters. Theoret. Comput. Sci. 410 (2009) 3893-3900. | MR | Zbl

[13] V. Keränen, Combinatorics on words - suppression of unfavorable factors in pattern avoidance. TMJ 11 (2010). Available at http://www.mathematica-journal.com/issue/v11i3/Keranen.html consulted in November 2011.

[14] M. Rao, Last cases of Dejean's conjecture. Theoret. Comput. Sci. 412 (2011) 3010-3018; Combinatorics on Words (WORDS 2009), 7th International Conference on Words. | MR | Zbl

[15] A.M. Shur, Comparing complexity functions of a language and its extendable part. RAIRO-Theor. Inf. Appl. 42 (2008) 647-655. | Numdam | MR | Zbl

[16] A. M. Shur, Growth rates of complexity of power-free languages. Theoret. Comput. Sci. 411 (2010) 3209-3223. | MR | Zbl

[17] A. Thue, Über unendliche Zeichenreihen. Kra. Vidensk. Selsk. Skrifter. I. Mat. Nat. Kl. Christiana 7 (1906) 1-22. | JFM

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