Hostname: page-component-78c5997874-j824f Total loading time: 0 Render date: 2024-11-11T09:14:03.009Z Has data issue: false hasContentIssue false
  • English
  • Français

Fewest repetitions in infinite binary words

Published online by Cambridge University Press:  26 August 2011

Golnaz Badkobeh  and
Maxime Crochemore
Get access

Abstract

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.

Keywords

Combinatorics on wordsrepetitionsword morphisms
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 46 , Issue 1: Special issue dedicated to the 13th "Journées Montoises d'Informatique Théorique" , January 2012 , pp. 17 - 31
Copyright
© EDP Sciences 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

G. Badkobeh and M. Crochemore, An infinite binary word containing only three distinct squares (2010) Submitted.
Currie, J.D. and Rampersad, N., A proof of Dejean’s conjecture. Math. Comput. 80 (2011) 1063–1070. Google Scholar
Dejean, F., Sur un théorème de Thue. J. Comb. Theory, Ser. A 13 (1972) 90–99. Google Scholar
A.S. Fraenkel and J. Simpson, How many squares must a binary sequence contain? Electr. J. Comb. 2 (1995).
Harju, T. and Nowotka, D., Binary words with few squares. Bulletin of the EATCS 89 (2006) 164–166. Google Scholar
Karhumäki, J. and Shallit, J., Polynomial versus exponential growth in repetition-free binary words. J. Comb. Th. (A) 105 (2004) 335–347. Google Scholar
M. Lothaire Ed., Combinatorics on Words. Cambridge University Press, 2nd edition (1997).
Rampersad, N., J. Shallit and M. Wei Wang, Avoiding large squares in infinite binary words. Theoret. Comput. Sci. 339 (2005) 19–34. Google Scholar
Rao, M., Last cases of Dejean’s conjecture. Theor. Comput. Sci. 412 (2011) 3010–3018. Google Scholar
Shallit, J., Simultaneous avoidance of large squares and fractional powers in infinite binary words. Int. J. Found. Comput. Sci. 15 (2004) 317–327. Google Scholar
Thue, A., Über unendliche Zeichenreihen. Norske Vid. Selsk. Skr. I. Mat. Nat. Kl. Christiana 7 (1906) 1–22. Google Scholar