On the product of balanced sequences

Antonio Restivo; Giovanna Rosone

RAIRO - Theoretical Informatics and Applications (2012)

  • Volume: 46, Issue: 1, page 131-145
  • ISSN: 0988-3754

Abstract

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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.

How to cite

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Restivo, Antonio, and Rosone, Giovanna. "On the product of balanced sequences." RAIRO - Theoretical Informatics and Applications 46.1 (2012): 131-145. <http://eudml.org/doc/221967>.

@article{Restivo2012,
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. },
author = {Restivo, Antonio, Rosone, Giovanna},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Infinite sequences; Sturmian words; balance; product; infinite sequences},
language = {eng},
month = {3},
number = {1},
pages = {131-145},
publisher = {EDP Sciences},
title = {On the product of balanced sequences},
url = {http://eudml.org/doc/221967},
volume = {46},
year = {2012},
}

TY - JOUR
AU - Restivo, Antonio
AU - Rosone, Giovanna
TI - On the product of balanced sequences
JO - RAIRO - Theoretical Informatics and Applications
DA - 2012/3//
PB - EDP Sciences
VL - 46
IS - 1
SP - 131
EP - 145
AB - 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.
LA - eng
KW - Infinite sequences; Sturmian words; balance; product; infinite sequences
UR - http://eudml.org/doc/221967
ER -

References

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  2. N. Chekhova, P. Hubert and A. Messaoudi, Propriétés combinatoires, ergodiques et arithmétiques de la substitution de Tribonacci. J. Théor. Nombres Bordeaux13 (2001) 371–394.  Zbl1038.37010
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  6. P. Hubert, Suites équilibrées (french). Theor. Comput. Sci.242 (2000) 91–108.  
  7. M. Lothaire, Algebraic Combinatorics on Words. Cambridge University Press (2002).  Zbl1001.68093
  8. M. Morse and G.A. Hedlund, Symbolic dynamics II. Sturmian trajectories. Amer. J. Math.62 (1940) 1–42.  Zbl0022.34003
  9. R.N. Risley and L.Q. Zamboni, A generalization of sturmian sequences : combinatorial structure and transcendence. Acta Arith.95 (2000) 167–184.  Zbl0953.11007
  10. P.V. Salimov, On uniform recurrence of a direct product. Discrete Math. Theoret. Comput. Sci.12 (2010) 1–8.  Zbl1286.68377
  11. L. Vuillon, Balanced words. Bull. Belg. Math. Soc.10 (2003) 787–805.  Zbl1070.68129

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