RAIRO-Theor. Inf. Appl.
Volume 46, Number 1, January-March 2012Special issue dedicated to the 13th "Journées Montoises d'Informatique Théorique"
|Page(s)||17 - 31|
|Published online||26 August 2011|
Fewest repetitions in infinite binary words
1 King’s College London, London, UK
2 King’s College London, London, UK and Université Paris-Est, France
Received: 2 November 2010
Accepted: 16 June 2011
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.
Mathematics Subject Classification: 68R15
Key words: Combinatorics on words / repetitions / word morphisms
© EDP Sciences 2011
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