The critical exponent of the Arshon words
Krieger, Dalia
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 44 (2010), p. 139-150 / Harvested from Numdam

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.

Publié le : 2010-01-01
DOI : https://doi.org/10.1051/ita/2010009
Classification:  68R15
@article{ITA_2010__44_1_139_0,
     author = {Krieger, Dalia},
     title = {The critical exponent of the Arshon words},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     volume = {44},
     year = {2010},
     pages = {139-150},
     doi = {10.1051/ita/2010009},
     mrnumber = {2604939},
     zbl = {1184.68375},
     language = {en},
     url = {http://dml.mathdoc.fr/item/ITA_2010__44_1_139_0}
}
Krieger, Dalia. The critical exponent of the Arshon words. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 44 (2010) pp. 139-150. doi : 10.1051/ita/2010009. http://gdmltest.u-ga.fr/item/ITA_2010__44_1_139_0/

[1] S.E. Arshon, 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.

[2] S.E. Arshon, A proof of the existence of infinite asymmetric sequences on n symbols. Mat. Sb. 2 (1937) 769-779 (in Russian, with French abstract). | JFM 63.0928.01 | Zbl 0018.11503

[3] J. Berstel, Mots sans carré et morphismes itérés. Discrete Math. 29 (1979) 235-244. | Zbl 0444.20050

[4] 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).

[5] J.D. Currie, No iterated morphism generates any Arshon sequence of odd order. Discrete Math. 259 (2002) 277-283. | Zbl 1011.68069

[6] S. Kitaev, Symbolic sequences, crucial words and iterations of a morphism. Ph.D. thesis, Göteborg, Sweden (2000).

[7] S. Kitaev, There are no iterative morphisms that define the Arshon sequence and the σ-sequence. J. Autom. Lang. Comb. 8 (2003) 43-50. | Zbl 1064.68053

[8] A.V. Klepinin and E.V. Sukhanov, On combinatorial properties of the Arshon sequence. Discrete Appl. Math. 114 (2001) 155-169. | Zbl 0995.68506

[9] P. Séébold, About some overlap-free morphisms on a n-letter alphabet. J. Autom. Lang. Comb. 7 (2002) 579-597. | Zbl 1095.68090

[10] P. Séébold, On some generalizations of the Thue-Morse morphism. Theoret. Comput. Sci. 292 (2003) 283-298. | Zbl 1064.68079

[11] A. Thue, Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. Mat. Nat. Kl. 1 (1912) 1-67. | JFM 44.0462.01

[12] N.Ya. Vilenkin, Formulas on cardboard. Priroda 6 (1991) 95-104 (in Russian). English summary available at http://www.ams.org/mathscinet/index.html, review no. MR1143732.