Combinatorial and Arithmetical Properties of Infinite Words Associated with Non-simple Quadratic Parry Numbers

Abstract

We study arithmetical and combinatorial properties of $β$-integers for $β$ being the root of the equation $x^2=mx-n, m,n \in \mathbb N, m \geq n+2\geq 3$. We determine with the accuracy of $\pm 1$ the maximal number of $β$-fractional positions, which may arise as a result of addition of two $β$-integers. For the infinite word $u_β$ coding distances between consecutive $β$-integers, we determine precisely also the balance. The word $u_β$ is the 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 $u_β$ is sturmian and therefore 1-balanced. On the simplest non-sturmian example with $n\geq 2$, we illustrate how closely the balance and arithmetical properties of $β$-integers are related.