Combinatorial and arithmetical properties of infinite words associated with non-simple quadratic Parry numbers
RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) no. 3, pp. 307-328.

We study some arithmetical and combinatorial properties of $\beta$-integers for $\beta$ being the larger root of the equation ${x}^{2}=mx-n,m,n\in ℕ,m\ge n+2\ge 3$. We determine with the accuracy of $±$ 1 the maximal number of $\beta$-fractional positions, which may arise as a result of addition of two $\beta$-integers. For the infinite word ${u}_{\beta }$ coding distances between the consecutive $\beta$-integers, we determine precisely also the balance. The word ${u}_{\beta }$ is the only fixed point of the morphism $A$ $\to$ ${A}^{m-1}B$ and $B$ $\to$ ${A}^{m-n-1}B$. In the case $n=1$, the corresponding infinite word ${u}_{\beta }$ is sturmian, and, therefore, $1$-balanced. On the simplest non-sturmian example with $n$ $\ge$ 2, we illustrate how closely the balance and the arithmetical properties of $\beta$-integers are related.

DOI : https://doi.org/10.1051/ita:2007025
Classification : 68R15,  11A63
Mots clés : balance property, arithmetics, beta-expansions, infinite words
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title = {Combinatorial and arithmetical properties of infinite words associated with non-simple quadratic {Parry} numbers},
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Balková, Lubomíra; Pelantová, Edita; Turek, Ondřej. Combinatorial and arithmetical properties of infinite words associated with non-simple quadratic Parry numbers. RAIRO - Theoretical Informatics and Applications - Informatique Théorique et Applications, Tome 41 (2007) no. 3, pp. 307-328. doi : 10.1051/ita:2007025. http://www.numdam.org/articles/10.1051/ita:2007025/

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