Least periods of factors of infinite words
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 43 (2009) no. 1, pp. 165-178.

We show that any positive integer is the least period of a factor of the Thue-Morse word. We also characterize the set of least periods of factors of a sturmian word. In particular, the corresponding set for the Fibonacci word is the set of Fibonacci numbers. As a by-product of our results, we give several new proofs and tightenings of well-known properties of sturmian words.

DOI: 10.1051/ita:2008006
Classification: 68R15
Keywords: periodicity, Fibonacci word, Thue-Morse word, sturmian word
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Currie, James D.; Saari, Kalle. Least periods of factors of infinite words. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 43 (2009) no. 1, pp. 165-178. doi : 10.1051/ita:2008006. http://www.numdam.org/articles/10.1051/ita:2008006/

[1] J.-P. Allouche and J. Shallit, The ubiquitous Prouhet-Thue-Morse sequence, in Sequences and Their Applications: Proceedings of SETA'98. Springer Series in Discrete Mathematics and Theoretical Computer Science, C. Ding, T. Helleseth and H. Niederreiter, Eds., Springer-Verlag, London (1999) 1-16. | MR | Zbl

[2] J. Berstel, On the index of Sturmian words. In Jewels are forever. Springer, Berlin (1999) 287-294. | MR | Zbl

[3] W.-T. Cao and Z.-Y. Wen, Some properties of the factors of Sturmian sequences. Theor. Comput. Sci. 304 (2003) 365-385. | MR | Zbl

[4] C. Choffrut and J. Karhumäki, Combinatorics on words. In A. Salomaa and G. Rozenberg, Eds., Handbook of Formal Languages, volume 1. Springer, Berlin (1997) 329-438. | MR

[5] L.J. Cummings, D.W. Moore and J. Karhumäki, Borders of Fibonacci strings. J. Comb. Math. Comb. Comput. 20 (1996) 81-87. | MR | Zbl

[6] D. Damanik and D. Lenz, Powers in Sturmian sequences. Eur. J. Combin. 24 (2003) 377-390. | MR | Zbl

[7] A. De Luca and A. De Luca, Some characterizations of finite Sturmian words. Theor. Comput. Sci. 356 (2006) 118-125. | MR | Zbl

[8] N.J. Fine and H.S. Wilf, Uniqueness theorems for periodic functions. Proc. Amer. Math. Soc. 16 (1965) 109-114. | MR | Zbl

[9] T. Harju and D. Nowotka, Minimal Duval extensions. Int. J. Found. Comput. Sci. 15 (2004) 349-354. | MR | Zbl

[10] M. Lothaire, Combinatorics on Words. Cambridge University Press, Cambridge (1997). | MR | Zbl

[11] M. Lothaire, Algebraic Combinatorics on Words, Encyclopedia of Mathematics and its Applications, Vol. 90. Cambridge University Press, Cambridge (2002). | MR | Zbl

[12] F. Mignosi and L.Q. Zamboni, A note on a conjecture of Duval and Sturmian words. RAIRO-Theor. Inf. Appl. 36 (2002) 1-3. | Numdam | MR | Zbl

[13] M. Mohammad-Noori and J.D. Currie, Dejean's conjecture and Sturmian words. Eur. J. Combin. 28 (2007) 876-890. | MR | Zbl

[14] K. Saari, Periods of factors of the Fibonacci word. in Proceedings of the Sixth International Conference on Words (WORDS'07). Institut de Mathématiques de Luminy (2007) 273-279.

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