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The critical exponent of the Arshon words

Published online by Cambridge University Press:  11 February 2010

Dalia Krieger
Dalia Krieger*
Affiliation:
Faculty of Mathematics and Computer Science, The Weizmann Institute of Science, POB 26, Rehovot 76100, Israel; dalia.krieger@gmail.com
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Abstract

Generalizing the results of Thue (for n = 2) [Norske Vid. Selsk. Skr. Mat. Nat. Kl. 1 (1912) 1–67] and of Klepinin and Sukhanov (for n = 3) [Discrete Appl. Math. 114 (2001) 155–169], we prove that for all n ≥ 2, the critical exponent of the Arshon word of order n is given by (3n–2)/(2n–2), and this exponent is attained at position 1.

Keywords

Arshon wordscritical exponent.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 44 , Issue 1: Special issue dedicated to the 12th "Journées Montoises d'Informatique Théorique" , January 2010 , pp. 139 - 150
Copyright
© EDP Sciences, 2010

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References

Arshon, S.E., A proof of the existence of infinite asymmetric sequences on n symbols. Matematicheskoe Prosveshchenie (Mathematical Education) 2 (1935) 24–33 (in Russian). Available electronically at http://ilib.mccme.ru/djvu/mp1/mp1-2.htm.
Arshon, S.E., A proof of the existence of infinite asymmetric sequences on n symbols. Mat. Sb. 2 (1937) 769–779 (in Russian, with French abstract).
Berstel, J., Mots sans carré et morphismes itérés. Discrete Math. 29 (1979) 235–244. CrossRef
J. Berstel, Axel Thue's papers on repetitions in words: a translation. Publications du Laboratoire de Combinatoire et d'Informatique Mathématique 20, Université du Québec à Montréal (1995).
Currie, J.D., No iterated morphism generates any Arshon sequence of odd order. Discrete Math. 259 (2002) 277–283. CrossRef
S. Kitaev, Symbolic sequences, crucial words and iterations of a morphism. Ph.D. thesis, Göteborg, Sweden (2000).
Kitaev, S., There are no iterative morphisms that define the Arshon sequence and the σ-sequence. J. Autom. Lang. Comb. 8 (2003) 43–50.
Klepinin, A.V. and Sukhanov, E.V., On combinatorial properties of the Arshon sequence. Discrete Appl. Math. 114 (2001) 155–169. CrossRef
Séébold, P., About some overlap-free morphisms on a n-letter alphabet. J. Autom. Lang. Comb. 7 (2002) 579–597.
Séébold, P., On some generalizations of the Thue–Morse morphism. Theoret. Comput. Sci. 292 (2003) 283–298. CrossRef
Thue, A., Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. Mat. Nat. Kl. 1 (1912) 1–67.
Vilenkin, N.Ya., Formulas on cardboard. Priroda 6 (1991) 95–104 (in Russian). English summary available at http://www.ams.org/mathscinet/index.html, review no. MR1143732.