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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

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