Complexity of infinite words associated with beta-expansions
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 38 (2004) no. 2, p. 163-185

We study the complexity of the infinite word u β associated with the Rényi expansion of 1 in an irrational base β>1. When β is the golden ratio, this is the well known Fibonacci word, which is sturmian, and of complexity (n)=n+1. For β such that d β (1)=t 1 t 2 t m is finite we provide a simple description of the structure of special factors of the word u β . When t m =1 we show that (n)=(m-1)n+1. In the cases when t 1 =t 2 ==t m-1 or t 1 >max{t 2 ,,t m-1 } we show that the first difference of the complexity function (n+1)-(n) takes value in {m-1,m} for every n, and consequently we determine the complexity of u β . We show that u β is an Arnoux-Rauzy sequence if and only if d β (1)=ttt1. On the example of β=1+2cos(2π/7), solution of X 3 =2X 2 +X-1, we illustrate that the structure of special factors is more complicated for d β (1) infinite eventually periodic. The complexity for this word is equal to 2n+1.

DOI : https://doi.org/10.1051/ita:2004009
Classification:  11A63,  11A67,  37B10,  68R15
Keywords: beta-expansions, complexity of infinite words
@article{ITA_2004__38_2_163_0,
     author = {Frougny, Christiane and Mas\'akov\'a, Zuzana and Pelantov\'a, Edita},
     title = {Complexity of infinite words associated with beta-expansions},
     journal = {RAIRO - Theoretical Informatics and Applications - Informatique Th\'eorique et Applications},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {2},
     year = {2004},
     pages = {163-185},
     doi = {10.1051/ita:2004009},
     zbl = {1104.11013},
     mrnumber = {2060775},
     language = {en},
     url = {http://www.numdam.org/item/ITA_2004__38_2_163_0}
}
Frougny, Christiane; Masáková, Zuzana; Pelantová, Edita. Complexity of infinite words associated with beta-expansions. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Volume 38 (2004) no. 2, pp. 163-185. doi : 10.1051/ita:2004009. http://www.numdam.org/item/ITA_2004__38_2_163_0/

[1] J.-P. Allouche, Sur la complexité des suites infinies. Bull. Belg. Math. Soc. Simon Stevin 1 (1994) 133-143. | MR 1318964 | Zbl 0803.68094

[2] P. Arnoux et G. Rauzy, Représentation géométrique de suites de complexité 2n+1. Bull. Soc. Math. France 119 (1991) 199-215. | Numdam | MR 1116845 | Zbl 0789.28011

[3] J. Berstel, Recent results on extensions of Sturmian words. J. Algebra Comput. 12 (2003) 371-385. | MR 1902372 | Zbl 1007.68141

[4] A. Bertrand, Développements en base de Pisot et répartition modulo 1. C. R. Acad. Sci. Paris 285A (1977) 419-421. | MR 447134 | Zbl 0362.10040

[5] A. Bertrand-Mathis, Comment écrire les nombres entiers dans une base qui n'est pas entière. Acta Math. Acad. Sci. Hungar. 54 (1989) 237-241. | Zbl 0695.10005

[6] J. Cassaigne, Complexité et facteurs spéciaux. Bull. Belg. Math. Soc. Simon Stevin 4 (1997) 67-88. | MR 1440670 | Zbl 0921.68065

[7] J. Cassaigne, S. Ferenczi and L. Zamboni, Imbalances in Arnoux-Rauzy sequences. Ann. Inst. Fourier 50 (2000) 1265-1276. | Numdam | MR 1799745 | Zbl 1004.37008

[8] S. Fabre, Substitutions et β-systèmes de numération. Theoret. Comput. Sci. 137 (1995) 219-236. | MR 1311222 | Zbl 0872.11017

[9] Ch. Frougny, J.-P. Gazeau and R. Krejcar, Additive and multiplicative properties of point sets based on beta-integers. Theoret. Comput. Sci. 303 (2003) 491-516. | MR 1990778 | Zbl 1036.11034

[10] M. Lothaire, Algebraic combinatorics on words. Cambridge University Press (2002). | MR 1905123 | Zbl 1001.68093

[11] W. Parry, On the β-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960) 401-416. | MR 142719 | Zbl 0099.28103

[12] J. Patera, Statistics of substitution sequences. On-line computer program, available at http://kmlinux.fjfi.cvut.cz/~patera/SubstWords.cgi

[13] A. Rényi, Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957) 477-493. | MR 97374 | Zbl 0079.08901

[14] K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc. 12 (1980) 269-278. | MR 576976 | Zbl 0494.10040

[15] W.P. Thurston, Groups, tilings, and finite state automata. Geometry supercomputer project research report GCG1, University of Minnesota (1989).

[16] O. Turek, Complexity and balances of the infinite word of β-integers for β=1+3, in Proc. of Words'03, Turku. TUCS Publication 27 (2003) 138-148. | Zbl 1040.68090