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

Balková, Lubomíra; Pelantová, Edita; Turek, Ondřej

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 $x^2=mx-n,m,n\in \mathbb{N},m\ge n+2\ge 3$. We determine with the accuracy of $\pm$ 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.

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