A note on constructing infinite binary words with polynomial subword complexity
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 47 (2013) no. 2, pp. 195-199.

Most of the constructions of infinite words having polynomial subword complexity are quite complicated, e.g., sequences of Toeplitz, sequences defined by billiards in the cube, etc. In this paper, we describe a simple method for constructing infinite words w over a binary alphabet  { a,b }  with polynomial subword complexity pw. Assuming w contains an infinite number of a's, our method is based on the gap function which gives the distances between consecutive b's. It is known that if the gap function is injective, we can obtain at most quadratic subword complexity, and if the gap function is blockwise injective, we can obtain at most cubic subword complexity. Here, we construct infinite binary words w such that pw(n) = Θ(nβ) for any real number β > 1.

DOI : 10.1051/ita/2013033
Classification : 68R15
Mots-clés : binary words, subword complexity, gap function 
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     author = {Blanchet-Sadri, Francine and Chen, Bob and Munteanu, Sinziana},
     title = {A note on constructing infinite binary words with polynomial subword complexity},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     pages = {195--199},
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Blanchet-Sadri, Francine; Chen, Bob; Munteanu, Sinziana. A note on constructing infinite binary words with polynomial subword complexity. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 47 (2013) no. 2, pp. 195-199. doi : 10.1051/ita/2013033. http://www.numdam.org/articles/10.1051/ita/2013033/

[1] J.-P. Allouche, Sur la complexité des suites infinies. Bull. Belgian Math. Soc. 1 (1994) 133-143. | MR | Zbl

[2] J.-P. Allouche and J. Shallit, Automatic Sequences: Theory, Applications, Generalizations. Cambridge University Press (2003). | MR | Zbl

[3] P. Arnoux, C. Mauduit, I. Shiokawa and J. Tamura, Complexity of sequences defined by billiards in the cube. Bull. Soc. Math. France 122 (1994) 1-12. | Numdam | MR | Zbl

[4] P. Arnoux, C. Mauduit, I. Shiokawa and J. Tamura, Rauzy's conjecture on billiards in the cube. Tokyo J. Math. 17 (1994) 211-218. | MR | Zbl

[5] P. Arnoux and G. Rauzy, Représentation géométrique de suites de complexité 2n + 1. Bull. Soc. Math. France 119 (1991) 199-215. | Numdam | MR | Zbl

[6] Y. Baryshnikov, Complexity of trajectories in rectangular billiards. Communicat. Math. Phys. 174 (1995) 43-56. | MR | Zbl

[7] N. Bedaride, Directional billiard complexity in rational polyhedra. Regular Chaotic Dyn. 3 (2003) 97-106. | MR | Zbl

[8] F. Blanchet-Sadri, A. Chakarov, L. Manuelli, J. Schwartz and S. Stich, Constructing partial words with subword complexities not achievable by full words. Theoret. Comput. Sci. 432 (2012) 21-27. | MR | Zbl

[9] J. Cassaigne, Complexité et facteurs spéciaux. Bull. Belg. Math. Soc. 4 (1997) 67-88. | MR | Zbl

[10] J. Cassaigne, Constructing infinite words of intermediate complexity. In DLT 2002, 6th International Conference on Developments in Language Theory, Kyoto, Japan. Springer-Verlag, Berlin. Lect. Notes Comput. Sci. 2450 (2003) 173-184. | MR | Zbl

[11] J. Cassaigne and J. Karhumäki, Toeplitz words, generalized periodicity and periodically iterated morphisms. European J. Combin. 18 (1997) 497-510. | MR | Zbl

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

[13] I. Gheorghiciuc, The subword complexity of a class of infinite binary words. Adv. Appl. Math. 39 (2007) 237-259. | MR | Zbl

[14] M. Koskas, Complexité des suites de Toeplitz. Discrete Math. 183 (1998) 161-183. | Zbl

[15] M. Lothaire, Algebraic Combinatorics on Words. Cambridge University Press, Cambridge (2002). | MR | Zbl

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

[17] M. Morse and G.A. Hedlund, Symbolic dynamics II. Sturmian trajectories. Amer. J. Math. 62 (1940) 1-42. | JFM | MR

[18] G. Rote, Sequences with subword complexity 2n. J. Number Theory 46 (1993) 196-213. | MR | Zbl

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