# 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 -

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## 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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