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.
Mots-clés : Arshon words, critical exponent
@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}, pages = {139--150}, publisher = {EDP-Sciences}, volume = {44}, number = {1}, year = {2010}, doi = {10.1051/ita/2010009}, mrnumber = {2604939}, zbl = {1184.68375}, language = {en}, url = {http://archive.numdam.org/articles/10.1051/ita/2010009/} }
TY - JOUR AU - Krieger, Dalia TI - The critical exponent of the Arshon words JO - RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications PY - 2010 SP - 139 EP - 150 VL - 44 IS - 1 PB - EDP-Sciences UR - http://archive.numdam.org/articles/10.1051/ita/2010009/ DO - 10.1051/ita/2010009 LA - en ID - ITA_2010__44_1_139_0 ER -
%0 Journal Article %A Krieger, Dalia %T The critical exponent of the Arshon words %J RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications %D 2010 %P 139-150 %V 44 %N 1 %I EDP-Sciences %U http://archive.numdam.org/articles/10.1051/ita/2010009/ %R 10.1051/ita/2010009 %G en %F 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) no. 1, pp. 139-150. doi : 10.1051/ita/2010009. http://archive.numdam.org/articles/10.1051/ita/2010009/
[1] 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] A proof of the existence of infinite asymmetric sequences on n symbols. Mat. Sb. 2 (1937) 769-779 (in Russian, with French abstract). | JFM | Zbl
,[3] Mots sans carré et morphismes itérés. Discrete Math. 29 (1979) 235-244. | Zbl
,[4] 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] No iterated morphism generates any Arshon sequence of odd order. Discrete Math. 259 (2002) 277-283. | Zbl
,[6] Symbolic sequences, crucial words and iterations of a morphism. Ph.D. thesis, Göteborg, Sweden (2000).
,[7] There are no iterative morphisms that define the Arshon sequence and the σ-sequence. J. Autom. Lang. Comb. 8 (2003) 43-50. | Zbl
,[8] On combinatorial properties of the Arshon sequence. Discrete Appl. Math. 114 (2001) 155-169. | Zbl
and ,[9] About some overlap-free morphisms on a n-letter alphabet. J. Autom. Lang. Comb. 7 (2002) 579-597. | Zbl
,[10] On some generalizations of the Thue-Morse morphism. Theoret. Comput. Sci. 292 (2003) 283-298. | Zbl
,[11] Über die gegenseitige Lage gleicher Teile gewisser Zeichenreihen. Norske Vid. Selsk. Skr. Mat. Nat. Kl. 1 (1912) 1-67. | JFM
,[12] Formulas on cardboard. Priroda 6 (1991) 95-104 (in Russian). English summary available at http://www.ams.org/mathscinet/index.html, review no. MR1143732.
,Cité par Sources :