Hereditary properties of words
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 39 (2005) no. 1, pp. 49-65.

Let 𝒫 be a hereditary property of words, i.e., an infinite class of finite words such that every subword (block) of a word belonging to 𝒫 is also in 𝒫. Extending the classical Morse-Hedlund theorem, we show that either 𝒫 contains at least n+1 words of length n for every n or, for some N, it contains at most N words of length n for every n. More importantly, we prove the following quantitative extension of this result: if 𝒫 has mn words of length n then, for every kn+m, it contains at most (m+1)/2(m+1)/2 words of length k.

DOI: 10.1051/ita:2005003
Classification: 05C
Keywords: graph properties, monotone, hereditary, speed, size
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Balogh, József; Bollobás, Béla. Hereditary properties of words. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 39 (2005) no. 1, pp. 49-65. doi : 10.1051/ita:2005003. http://www.numdam.org/articles/10.1051/ita:2005003/

[1] S. Ferenczi, Rank and symbolic complexity. Ergodic Theory Dyn. Syst. 16 (1996) 663-682. | Zbl

[2] S. Ferenczi, Complexity of sequences and dynamical systems. Discrete Math. 206 (1999) 145-154. | Zbl

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

[4] A. Heinis, The P(n)/n-function for bi-infinite words. Theoret. Comput. Sci. 273 (2002) 35-46. | Zbl

[5] T. Kamae and L. Zamboni, Sequence entropy and the maximal pattern complexity of infinite words. Ergodic Theory Dynam. Syst. 22 (2002) 1191-1199. | Zbl

[6] M. Morse and A.G. Hedlund, Symbolic dynamics. Amer. J. Math 60 (1938) 815-866. | JFM

[7] R. Tijdeman, Periodicity and almost periodicity | MR | Zbl

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