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Combinatorial and arithmetical properties of infinite words associated with non-simple quadratic Parry numbers

Published online by Cambridge University Press:  25 September 2007

Lubomíra Balková ,
Edita Pelantová  and
Ondřej Turek
Lubomíra Balková
Affiliation:
Doppler Institute for Mathematical Physics and Applied Mathematics and Department of Mathematics, FNSPE, Czech Technical University, Trojanova 13, 120 00 Praha 2, Czech Republic; [email protected]; [email protected]; [email protected]
Edita Pelantová
Affiliation:
Doppler Institute for Mathematical Physics and Applied Mathematics and Department of Mathematics, FNSPE, Czech Technical University, Trojanova 13, 120 00 Praha 2, Czech Republic; [email protected]; [email protected]; [email protected]
Ondřej Turek
Affiliation:
Doppler Institute for Mathematical Physics and Applied Mathematics and Department of Mathematics, FNSPE, Czech Technical University, Trojanova 13, 120 00 Praha 2, Czech Republic; [email protected]; [email protected]; [email protected]
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Abstract

We study some arithmetical and combinatorial properties of β-integers for β being the larger root of the equation x2 = mx - n,m,n ∈ ℵ, m ≥ n +2 ≥ 3. 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 uβ> coding distances between the consecutive β-integers, we determine precisely also the balance. The word uβ> is the only fixed point of the morphism A → Am-1B and B → Am-n-1B. In the case n = 1, the corresponding infinite word uβ> 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.

Keywords

Balance propertyarithmeticsbeta-expansionsinfinite words
Type
Research Article
Information
RAIRO - Theoretical Informatics and Applications , Volume 41 , Issue 3 , July 2007 , pp. 307 - 328
Copyright
© EDP Sciences, 2007

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References

Adamczewski, B., Balances for fixed points of primitive substitutions. Theoret. Comput. Sci. 307 (2003) 47–75. CrossRef
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.
Ambrož, P., Frougny, Ch., Masáková, Z. and Pelantová, E., Arithmetics on number systems with irrational bases. Bull. Belg. Math. Soc. Simon Stevin 10 (2003) 641–659.
Ambrož, P., Frougny, Ch., Masáková, Z. and Pelantová, E., Palindromic complexity of infinite words associated with simple Parry numbers. Ann. Institut Fourier 56 (2006) 2131–2160. CrossRef
Ambrož, P., Masáková, Z. and Pelantová, E., Addition and multiplication of beta-expansions in generalized Tribonacci base. Discrete Math. Theor. Comput. Sci. 9 (2007) 73-88.
Bernat, J., Computation of L ⊕ for several cubic Pisot numbers. Discrete Math. Theor. Comput. Sci. 9 (2007) 175-194.
Berstel, J., Recent results on extension of sturmian words. Int. J. Algebr. Comput. 12 (2002) 371–385. CrossRef
Bertrand, A., Développements en base de Pisot et répartition modulo 1. C. R. Acad. Sci. Paris 285 (1977) 419–421.
Burdík, Č., Frougny, Ch., Gazeau, J.P. and Krejcar, R., Beta-integers as natural counting systems for quasicrystals. J. Phys. A 31 (1998) 6449–6472. CrossRef
Fabre, S., Substitutions et β-systèmes de numération. Theoret. Comput. Sci. 137 (1995) 219–236. CrossRef
Ch. Frougny, B. Solomyak, Finite β-expansions. Ergodic Theory Dynam. Systems 12 (1994) 713–723.
Ch. Frougny, Z. Masáková, 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. CrossRef
Ch. Frougny, Z. Masáková, E. Pelantová, Infinite special branches in words associated with beta-expansions. Discrete Math. Theor. Comput. Sci. 9 (2007) 125-144.
L.S. Guimond, Z. Masáková and E. Pelantová, Arithmetics of β-expansions, Acta Arithmetica 112 (2004) 23–40. CrossRef
M. Hollander, Linear numeration systems, finite beta-expansions, and discrete spectrum of substitution dynamical systems. Ph.D. Thesis, Washington University, USA (1996)
Justin, J. and Pirillo, G., On a combinatorial property of sturmian words. Theoret. Comput. Sci. 154 (1996) 387–394. CrossRef
Lagarias, J., Geometric models for quasicrystals I. Delone sets of finite type. Discrete Comput. Geom. 21 (1999) 161–191. CrossRef
Messaoudi, A., Généralisation de la multiplication de Fibonacci. Math. Slovaca 50 (2000) 135–148.
Y. Meyer. Quasicrystals, Diophantine approximation, and algebraic numbers, in Beyond Quasicrystals, edited by F. Axel and D. Gratias. Springer (1995) 3–16.
Morse, M. and Hedlund, G.A., Symbolic dynamics II. Sturmian trajectories. Amer. J. Math. 62 (1940) 1–42. CrossRef
Parry, W., On the beta-expansions of real numbers. Acta Math. Acad. Sci. Hungar. 11 (1960) 401–416. CrossRef
Rényi, A., Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar. 8 (1957) 477–493. CrossRef
Schmidt, K., On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc. 12 (1980) 269–278. CrossRef
Shechtman, D., Blech, I., Gratias, D. and Cahn, J., Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53 (1984) 1951–1954. CrossRef
W.P. Thurston, Groups, tilings, and finite state automata, Geometry supercomputer project research report GCG1, University of Minnesota, USA (1989).
Turek, O., Balance properties of the fixed point of the substitution associated to quadratic simple Pisot numbers. RAIRO-Theor. Inf. Appl. 41 (2007) 123–135. CrossRef