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

Abstract

We study arithmetical and combinatorial properties of $\beta$-integers for $\beta$ being the 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 consecutive $\beta$-integers, we determine precisely also the balance. The word $u_{\beta }$ 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 $u_{\beta }$ is sturmian and therefore 1-balanced. On the simplest non-sturmian example with $n\ge 2$, we illustrate how closely the balance and arithmetical properties of $\beta$-integers are related.

15 pages