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

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

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## Abstract

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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

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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 -

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