Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-26T17:39:40.463Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access

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

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

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.