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Sequences of low arithmetical complexity

Published online by Cambridge University Press:  08 November 2006

Sergey V. Avgustinovich ,
Julien Cassaigne  and
Anna E. Frid
Sergey V. Avgustinovich
Affiliation:
Sobolev Institute of Mathematics SB RAS, Koptyug Av. 4, Novosibirsk, Russia; [email protected]; [email protected]
Julien Cassaigne
Affiliation:
Institut de Mathématiques de Luminy, case 907, 163 Av. de Luminy, 13288 Marseille Cedex 9, France; [email protected]
Anna E. Frid
Affiliation:
Sobolev Institute of Mathematics SB RAS, Koptyug Av. 4, Novosibirsk, Russia; [email protected]; [email protected]
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Abstract

Arithmetical complexity of a sequence is the number of words of length n that can be extracted from it according to arithmetic progressions. We study uniformly recurrent words of low arithmetical complexity and describe the family of such words having lowest complexity.

Keywords

Arithmetical complexityinfinite wordsToeplitz wordsspecialfactorsperiod doubling wordLegendre symbolpaperfolding word.
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 40 , Issue 4 , October 2006 , pp. 569 - 582
Copyright
© EDP Sciences, 2006

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References

Allouche, J.-P., The number of factors in a paperfolding sequence. Bull. Austral. Math. Soc. 46 (1992) 23–32. CrossRef
Allouche, J.-P., Baake, M., Cassaigne, J. and Damanik, D., Palindrome complexity. Theoret. Comput. Sci. 292 (2003) 9–31. CrossRef
S.V. Avgustinovich, D.G. Fon-Der-Flaass and A.E. Frid, Arithmetical complexity of infinite words, in Words, Languages & Combinatorics III, edited by M. Ito and T. Imaoka. Singapore, World Scientific Publishing, ICWLC 2000, Kyoto, Japan, March 14–18 (2003) 51–62.
J. Berstel and P. Séébold, Sturmian words, in Algebraic combinatorics on words, edited by M. Lothaire. Cambridge University Press (2002).
A.A. Bukhshtab, Number Theory. Uchpedgiz, Moscow (1960) (in Russian).
Cassaigne, J., Complexité et facteurs spéciaux. Bull. Belg. Math. Soc. Simon Stevin 4 (1997) 67–88.
J. Cassaigne and A. Frid, On arithmetical complexity of Sturmian words, in Proc. WORDS 2005, Montreal (2005) 197–208.
J. Cassaigne and J. Karhumäki, Toeplitz words, generalized periodicity and periodically iterated morphisms. Eur. J. Combin. 18 (1997) 497–510.
Damanik, D., Local symmetries in the period doubling sequence. Discrete Appl. Math. 100 (2000) 115–121. CrossRef
Ferenczi, S., Complexity of sequences and dynamical systems. Discrete Math. 206 (1999) 145–154. CrossRef
A. Frid, A lower bound for the arithmetical complexity of Sturmian words, Siberian Electronic Mathematical Reports 2, 14–22 [Russian, English abstract].
Frid, A., Arithmetical complexity of symmetric D0L words. Theoret. Comput. Sci. 306 (2003) 535–542. CrossRef
Frid, A., Possible Growth, On of Arithmetical Complexity. RAIRO-Inf. Theor. Appl. 40 (2006) 443–458. CrossRef
Frid, A., Sequences of linear arithmetical complexity. Theoret. Comput. Sci. 339 (2005) 68–87. CrossRef
Justin, J. and Pirillo, G., Decimations and Sturmian words. Theor. Inform. Appl. 31 (1997) 271–290. CrossRef
Kamae, T. and Zamboni, L., Maximal pattern complexity for discrete systems. Ergodic Theory Dynam. Syst. 22 (2002) 1201–1214.
Koskas, M., Complexités de suites de Toeplitz. Discrete Math. 183 (1998) 161–183. CrossRef
Nakashima, I., Tamura, J., Yasutomi, S., Nakashima, I., Tamura, J.-I. and Yasutomi, S.-I., *-Sturmian words and complexity. J. Théor. Nombres Bordeaux 15 (2003) 767–804. CrossRef
A. Restivo and S. Salemi, Binary patterns in infinite binary words, in Formal and Natural Computing, edited by W. Brauer et al. Lect. Notes Comput. Sci. 2300, (2002) 107–116.
E. Szemerédi, On sets of integers containing no k elements in arithmetic progression. Acta Arith. 27 (1975) 199–245.