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Finite-Repetition threshold for infinite ternary words

Golnaz Badkobeh
(King's College London)
Maxime Crochemore
(King's College London)

The exponent of a word is the ratio of its length over its smallest period. The repetitive threshold r(a) of an a-letter alphabet is the smallest rational number for which there exists an infinite word whose finite factors have exponent at most r(a). This notion was introduced in 1972 by Dejean who gave the exact values of r(a) for every alphabet size a as it has been eventually proved in 2009.

The finite-repetition threshold for an a-letter alphabet refines the above notion. It is the smallest rational number FRt(a) for which there exists an infinite word whose finite factors have exponent at most FRt(a) and that contains a finite number of factors with exponent r(a). It is known from Shallit (2008) that FRt(2)=7/3.

With each finite-repetition threshold is associated the smallest number of r(a)-exponent factors that can be found in the corresponding infinite word. It has been proved by Badkobeh and Crochemore (2010) that this number is 12 for infinite binary words whose maximal exponent is 7/3.

We show that FRt(3)=r(3)=7/4 and that the bound is achieved with an infinite word containing only two 7/4-exponent words, the smallest number.

Based on deep experiments we conjecture that FRt(4)=r(4)=7/5. The question remains open for alphabets with more than four letters.

Keywords: combinatorics on words, repetition, repeat, word powers,

word exponent, repetition threshold, pattern avoidability, word morphisms.

In Petr Ambrož, Štěpán Holub and Zuzana Masáková: Proceedings 8th International Conference Words 2011 (WORDS 2011), Prague, Czech Republic, 12-16th September 2011, Electronic Proceedings in Theoretical Computer Science 63, pp. 37–43.
Published: 17th August 2011.

ArXived at: http://dx.doi.org/10.4204/EPTCS.63.7 bibtex PDF
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