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

Lubomíra Balkova

Doppler Institute for Mathematical Physics and Applied Mathematics and Department of Mathematics, FNSPE, Czech Technical University, Prag, Czech Republic

Coauthor(s): Edita Pelantová and Ondrej Turek

We study some arithmetical and combinatorial properties of $\beta$-integers for $\beta$ being the larger root of the equation $x^2=mx-n, m,n \in \mathbb{N}, m \geq n+2\geq 3$. In order to define $\beta$-integers, we first need to know that $\beta$-expansion of a real number $x$ in base $\beta >1$ is its lexicographically largest expansion in this base. A real number $x$ is then called a $\beta$-integer if its $\beta$-expansion has the form $\pm \sum_{k=0}^{n} x_k \beta^{k}$, i.e., if all of its coefficients at powers $\beta^{k}$ vanish for $k<0$. 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. The infinite word $u_\beta$ coding distances between the consecutive $\beta$-integers 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. We determine precisely also the balance for non-sturmian cases; the difference in numbers of different letters in factors of the same length is at most $\left\lceil\frac{m-2}{m-n-1}\right\rceil.$ On the simplest non-sturmian example with $n\geq 2$, we illustrate how closely the balance and the arithmetical properties of $\beta$-integers are related.