Combinatorial and arithmetical properties of infinite words associated with non-simple quadratic Parry numbers
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) no. 3, pp. 307-328.

We study some arithmetical and combinatorial properties of β-integers for β being the larger root of the equation x 2 =mx-n,m,n,mn+23. We determine with the accuracy of ± 1 the maximal number of β-fractional positions, which may arise as a result of addition of two β-integers. For the infinite word u β coding distances between the consecutive β-integers, we determine precisely also the balance. The word u β is the only fixed point of the morphism A A m-1 B and B A m-n-1 B. In the case n=1, the corresponding infinite word u β is sturmian, and, therefore, 1-balanced. On the simplest non-sturmian example with n 2, we illustrate how closely the balance and the arithmetical properties of β-integers are related.

DOI : 10.1051/ita:2007025
Classification : 68R15, 11A63
Mots clés : balance property, arithmetics, beta-expansions, infinite words
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     title = {Combinatorial and arithmetical properties of infinite words associated with non-simple quadratic {Parry} numbers},
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Balková, Lubomíra; Pelantová, Edita; Turek, Ondřej. Combinatorial and arithmetical properties of infinite words associated with non-simple quadratic Parry numbers. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) no. 3, pp. 307-328. doi : 10.1051/ita:2007025. http://archive.numdam.org/articles/10.1051/ita:2007025/

[1] B. Adamczewski, Balances for fixed points of primitive substitutions. Theoret. Comput. Sci. 307 (2003) 47-75. | Zbl

[2] S. Akiyama, Cubic Pisot units with finite beta expansions, in Algebraic Number Theory and Diophantine Analysis, edited by F. Halter-Koch and R.F. Tichy. De Gruyter, Berlin (2000) 11-26. | Zbl

[3] P. Ambrož, Ch. Frougny, Z. Masáková and E. Pelantová, Arithmetics on number systems with irrational bases. Bull. Belg. Math. Soc. Simon Stevin 10 (2003) 641-659. | Zbl

[4] P. Ambrož, Ch. Frougny, Z. Masáková and E. Pelantová, Palindromic complexity of infinite words associated with simple Parry numbers. Ann. Institut Fourier 56 (2006) 2131-2160. | Numdam | Zbl

[5] P. Ambrož, Z. Masáková and E. Pelantová, Addition and multiplication of beta-expansions in generalized Tribonacci base. Discrete Math. Theor. Comput. Sci. 9 (2007) 73-88. | MR

[6] J. Bernat, Computation of L for several cubic Pisot numbers. Discrete Math. Theor. Comput. Sci. 9 (2007) 175-194. | MR

[7] J. Berstel, Recent results on extension of sturmian words. Int. J. Algebr. Comput. 12 (2002) 371-385. | Zbl

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

[9] Č. Burdík, Ch. Frougny, J.P. Gazeau and R. Krejcar, Beta-integers as natural counting systems for quasicrystals. J. Phys. A 31 (1998) 6449-6472. | Zbl

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

[11] Ch. Frougny and B. Solomyak, Finite β-expansions. Ergodic Theory Dynam. Systems 12 (1994) 713-723. | Zbl

[12] Ch. Frougny, Z. Masáková and E. Pelantová, Complexity of infinite words associated with beta-expansions. RAIRO-Theor. Inf. Appl. 38 (2004) 163-185; Corrigendum, RAIRO-Theor. Inf. Appl. 38 (2004) 269-271. | EuDML | Numdam | Zbl

[13] Ch. Frougny, Z. Masáková and E. Pelantová, Infinite special branches in words associated with beta-expansions. Discrete Math. Theor. Comput. Sci. 9 (2007) 125-144. | MR | Zbl

[14] L.S. Guimond, Z. Masáková and E. Pelantová, Arithmetics of β-expansions, Acta Arithmetica 112 (2004) 23-40. | EuDML | Zbl

[15] M. Hollander, Linear numeration systems, finite beta-expansions, and discrete spectrum of substitution dynamical systems. Ph.D. Thesis, Washington University, USA (1996)

[16] J. Justin and G. Pirillo, On a combinatorial property of sturmian words. Theoret. Comput. Sci. 154 (1996) 387-394. | Zbl

[17] J. Lagarias, Geometric models for quasicrystals I. Delone sets of finite type. Discrete Comput. Geom. 21 (1999) 161-191. | Zbl

[18] A. Messaoudi, Généralisation de la multiplication de Fibonacci. Math. Slovaca 50 (2000) 135-148. | EuDML | Zbl

[19] Y. Meyer. Quasicrystals, Diophantine approximation, and algebraic numbers, in Beyond Quasicrystals, edited by F. Axel and D. Gratias. Springer (1995) 3-16. | Zbl

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

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

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

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

[24] D. Shechtman, I. Blech, D. Gratias and J. Cahn, Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53 (1984) 1951-1954.

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

[26] O. Turek, Balance properties of the fixed point of the substitution associated to quadratic simple Pisot numbers. RAIRO-Theor. Inf. Appl. 41 (2007) 123-135. | EuDML | Numdam | Zbl

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