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On the product of balanced sequences

Published online by Cambridge University Press:  14 September 2011

Antonio Restivo  and
Giovanna Rosone
Antonio Restivo
Affiliation:
University of Palermo, Dipartimento di Matematica e Informatica, Via Archirafi 34, 90123 Palermo, Italy. restivo@math.unipa.it, giovanna@math.unipa.it
Giovanna Rosone
Affiliation:
University of Palermo, Dipartimento di Matematica e Informatica, Via Archirafi 34, 90123 Palermo, Italy. restivo@math.unipa.it, giovanna@math.unipa.it
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Abstract

The product w = u ⊗ v of two sequences u and v is a naturally defined sequence on the alphabet of pairs of symbols. Here, we study when the product w of two balanced sequences u,v is balanced too. In the case u and v are binary sequences, we prove, as a main result, that, if such a product w is balanced and deg(w) = 4, then w is an ultimately periodic sequence of a very special form. The case of arbitrary alphabets is approached in the last section. The partial results obtained and the problems proposed show the interest of the notion of product in the study of balanced sequences.

Keywords

Infinite sequencesSturmian wordsbalanceproduct
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. 131 - 145
Copyright
© EDP Sciences 2011

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References

Références

Altman, E., Gaujal, B. and Hordijk, A., Balanced sequences and optimal routing. J. ACM 47 (2000) 752–775. Google Scholar
Chekhova, N., Hubert, P. and Messaoudi, A., Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci. J. Théor. Nombres Bordeaux 13 (2001) 371–394. Google Scholar
C. Choffrut and J. Karhumaki, Combinatorics of words, in G. Rozenberg and A. Salomaa eds., Handbook of Formal Language Theory 1. Springer-Verlag, Berlin (1997).
Ferenczi, S. and Mauduit, C., Transcendence of numbers with a low complexity expansion. J. Number Theory 67 (1997) 146–161. Google Scholar
Fraenkel, A.S., Complementing and exactly covering sequences. J. Combin. Theory Ser. A 14 (1973) 8–20. Google Scholar
Hubert, P., Suites équilibrées (french). Theor. Comput. Sci. 242 (2000) 91–108. Google Scholar
M. Lothaire, Algebraic Combinatorics on Words. Cambridge University Press (2002).
Morse, M. and Hedlund, G.A., Symbolic dynamics II. Sturmian trajectories. Amer. J. Math. 62 (1940) 1–42. Google Scholar
Risley, R.N. and Zamboni, L.Q., A generalization of sturmian sequences : combinatorial structure and transcendence. Acta Arith. 95 (2000) 167–184. Google Scholar
Salimov, P.V., On uniform recurrence of a direct product. Discrete Math. Theoret. Comput. Sci. 12 (2010) 1–8. Google Scholar
Vuillon, L., Balanced words. Bull. Belg. Math. Soc. 10 (2003) 787–805. Google Scholar