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word morphisms

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Fewest repetitions in infinite binary words
Golnaz Badkobeh, Maxime Crochemore
Journal:

RAIRO - Theoretical Informatics and Applications / Volume 46 / Issue 1 / January 2012

Published online by Cambridge University Press:

26 August 2011, pp. 17-31

Print publication:

January 2012
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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.