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

Lubomíra Balková; Edita Pelantová; Ondřej Turek

RAIRO - Theoretical Informatics and Applications (2007)

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

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topBalková, 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- B. Adamczewski, Balances for fixed points of primitive substitutions. Theoret. Comput. Sci.307 (2003) 47–75.
- 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.
- 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.
- 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.
- 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.
- J. Bernat, Computation of L⊕ for several cubic Pisot numbers. Discrete Math. Theor. Comput. Sci.9 (2007) 175-194.
- J. Berstel, Recent results on extension of sturmian words. Int. J. Algebr. Comput.12 (2002) 371–385.
- A. Bertrand, Développements en base de Pisot et répartition modulo 1. C. R. Acad. Sci. Paris285 (1977) 419–421.
- Č. Burdík, Ch. Frougny, J.P. Gazeau and R. Krejcar, Beta-integers as natural counting systems for quasicrystals. J. Phys. A31 (1998) 6449–6472.
- S. Fabre, Substitutions et β-systèmes de numération. Theoret. Comput. Sci.137 (1995) 219–236.
- Ch. Frougny and B. Solomyak, Finite β-expansions. Ergodic Theory Dynam. Systems12 (1994) 713–723.
- 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.
- 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.
- L.S. Guimond, Z. Masáková and E. Pelantová, Arithmetics of β-expansions, Acta Arithmetica112 (2004) 23–40.
- M. Hollander, Linear numeration systems, finite beta-expansions, and discrete spectrum of substitution dynamical systems. Ph.D. Thesis, Washington University, USA (1996)
- J. Justin and G. Pirillo, On a combinatorial property of sturmian words. Theoret. Comput. Sci.154 (1996) 387–394.
- J. Lagarias, Geometric models for quasicrystals I. Delone sets of finite type. Discrete Comput. Geom.21 (1999) 161–191.
- A. Messaoudi, Généralisation de la multiplication de Fibonacci. Math. Slovaca50 (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.
- M. Morse and G.A. Hedlund, Symbolic dynamics II. Sturmian trajectories. Amer. J. Math.62 (1940) 1–42.
- W. Parry, On the beta-expansions of real numbers. Acta Math. Acad. Sci. Hungar.11 (1960) 401–416.
- A. Rényi, Representations for real numbers and their ergodic properties. Acta Math. Acad. Sci. Hungar.8 (1957) 477–493.
- K. Schmidt, On periodic expansions of Pisot numbers and Salem numbers. Bull. London Math. Soc.12 (1980) 269–278.
- 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.
- W.P. Thurston, Groups, tilings, and finite state automata, Geometry supercomputer project research report GCG1, University of Minnesota, USA (1989).
- 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.

## Citations in EuDML Documents

top- L'ubomíra Balková, Zuzana Masáková, Palindromic complexity of infinite words associated with non-simple Parry numbers
- L'ubomíra Balková, Zuzana Masáková, Palindromic complexity of infinite words associated with non-simple Parry numbers
- Ondřej Turek, Balances and Abelian Complexity of a Certain Class of Infinite Ternary Words
- Z. Masáková, T. Vávra, Integers in number systems with positive and negative quadratic Pisot base
- L'ubomíra Balková, Edita Pelantová, Štěpán Starosta, Sturmian jungle (or garden?) on multiliteral alphabets
- L'ubomíra Balková, Edita Pelantová, Štěpán Starosta, Sturmian jungle (or garden?) on multiliteral alphabets

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