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

• Volume: 41, Issue: 3, page 307-328
• ISSN: 0988-3754

top

Abstract

top
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.

How to cite

top

Balková, Lubomíra, Pelantová, Edita, and Turek, Ondřej. "Combinatorial and arithmetical properties of infinite words associated with non-simple quadratic Parry numbers." RAIRO - Theoretical Informatics and Applications 41.3 (2007): 307-328. <http://eudml.org/doc/250037>.

@article{Balková2007,
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. },
author = {Balková, Lubomíra, Pelantová, Edita, Turek, Ondřej},
journal = {RAIRO - Theoretical Informatics and Applications},
keywords = {Balance property; arithmetics; beta-expansions; infinite words; balance properties; Sturmian sequences; combinatorics on words},
language = {eng},
month = {9},
number = {3},
pages = {307-328},
publisher = {EDP Sciences},
title = {Combinatorial and arithmetical properties of infinite words associated with non-simple quadratic Parry numbers},
url = {http://eudml.org/doc/250037},
volume = {41},
year = {2007},
}

TY - JOUR
AU - Balková, Lubomíra
AU - Pelantová, Edita
AU - Turek, Ondřej
TI - Combinatorial and arithmetical properties of infinite words associated with non-simple quadratic Parry numbers
JO - RAIRO - Theoretical Informatics and Applications
DA - 2007/9//
PB - EDP Sciences
VL - 41
IS - 3
SP - 307
EP - 328
AB - 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.
LA - eng
KW - Balance property; arithmetics; beta-expansions; infinite words; balance properties; Sturmian sequences; combinatorics on words
UR - http://eudml.org/doc/250037
ER -

References

top
1. B. Adamczewski, Balances for fixed points of primitive substitutions. Theoret. Comput. Sci.307 (2003) 47–75.
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.
3. P. Ambrož, Ch. Frougny, Z. Masáková and E. Pelantová, Arithmetics on number systems with irrational bases. Bull. Belg. Math. Soc. Simon Stevin10 (2003) 641–659.
4. P. Ambrož, Ch. Frougny, Z. Masáková and E. Pelantová, Palindromic complexity of infinite words associated with simple Parry numbers. Ann. Institut Fourier56 (2006) 2131–2160.
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.
6. J. Bernat, Computation of L⊕ for several cubic Pisot numbers. Discrete Math. Theor. Comput. Sci.9 (2007) 175-194.
7. J. Berstel, Recent results on extension of sturmian words. Int. J. Algebr. Comput.12 (2002) 371–385.
8. A. Bertrand, Développements en base de Pisot et répartition modulo 1. C. R. Acad. Sci. Paris285 (1977) 419–421.
9. Č. Burdík, Ch. Frougny, J.P. Gazeau and R. Krejcar, Beta-integers as natural counting systems for quasicrystals. J. Phys. A31 (1998) 6449–6472.
10. S. Fabre, Substitutions et β-systèmes de numération. Theoret. Comput. Sci.137 (1995) 219–236.
11. Ch. Frougny and B. Solomyak, Finite β-expansions. Ergodic Theory Dynam. Systems12 (1994) 713–723.
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.
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.
14. L.S. Guimond, Z. Masáková and E. Pelantová, Arithmetics of β-expansions, Acta Arithmetica112 (2004) 23–40.
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.
17. J. Lagarias, Geometric models for quasicrystals I. Delone sets of finite type. Discrete Comput. Geom.21 (1999) 161–191.
18. A. Messaoudi, Généralisation de la multiplication de Fibonacci. Math. Slovaca50 (2000) 135–148.
19. Y. Meyer. Quasicrystals, Diophantine approximation, and algebraic numbers, in Beyond Quasicrystals, edited by F. Axel and D. Gratias. Springer (1995) 3–16.
20. M. Morse and G.A. Hedlund, Symbolic dynamics II. Sturmian trajectories. Amer. J. Math.62 (1940) 1–42.
21. W. Parry, On the beta-expansions of real numbers. Acta Math. Acad. Sci. Hungar.11 (1960) 401–416.
22. A. Rényi, Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar.8 (1957) 477–493.
23. K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc.12 (1980) 269–278.
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.

top

NotesEmbed?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.