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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; l.balkova@centrum.cz; masakova@km1.fjfi.cvut.cz; oturek@centrum.cz
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; l.balkova@centrum.cz; masakova@km1.fjfi.cvut.cz; oturek@centrum.cz
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; l.balkova@centrum.cz; masakova@km1.fjfi.cvut.cz; oturek@centrum.cz
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Abstract

We study some arithmetical and combinatorial properties ofβ-integers for β being the larger root of the equationx2 = mx - n,m,n ∈ ℵ, m ≥ n +2 ≥ 3. We determine withthe accuracy of ± 1 the maximal number of β-fractionalpositions, which may arise as a result of addition of twoβ-integers. For the infinite word uβ> coding distancesbetween the consecutive β-integers, we determine preciselyalso the balance. The word uβ> is the only fixed point of themorphism 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 withn≥ 2, we illustrate how closely the balance and thearithmetical properties of β-integers are related.

Keywords

Balance propertyarithmeticsbeta-expansionsinfinite words

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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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