Fewest repetitions in infinite binary words
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 46 (2012) no. 1, pp. 17-31.

A square is the concatenation of a nonempty word with itself. A word has period p if its letters at distance p match. The exponent of a nonempty word is the quotient of its length over its smallest period. In this article we give a proof of the fact that there exists an infinite binary word which contains finitely many squares and simultaneously avoids words of exponent larger than 7/3. Our infinite word contains 12 squares, which is the smallest possible number of squares to get the property, and 2 factors of exponent 7/3. These are the only factors of exponent larger than 2. The value 7/3 introduces what we call the finite-repetition threshold of the binary alphabet. We conjecture it is 7/4 for the ternary alphabet, like its repetitive threshold.

DOI: 10.1051/ita/2011109
Classification: 68R15
Keywords: combinatorics on words, repetitions, word morphisms
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Badkobeh, Golnaz; Crochemore, Maxime. Fewest repetitions in infinite binary words. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 46 (2012) no. 1, pp. 17-31. doi : 10.1051/ita/2011109. http://www.numdam.org/articles/10.1051/ita/2011109/

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