Combinatorial and Arithmetical Properties of Infinite Words Associated with Non-simple Quadratic Parry Numbers
<|reference_start|>Combinatorial and Arithmetical Properties of Infinite Words Associated with Non-simple Quadratic Parry Numbers: We study arithmetical and combinatorial properties of $\beta$-integers for $\beta$ being the root of the equation $x^2=mx-n, m,n \in \mathbb N, m \geq n+2\geq 3$. 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 $n\geq 2$, we illustrate how closely the balance and arithmetical properties of $\beta$-integers are related.<|reference_end|>
@article{balková2006combinatorial,
    title={Combinatorial and Arithmetical Properties of Infinite Words Associated
  with Non-simple Quadratic Parry Numbers},
    author={Lubom'ira Balkov'a, Edita Pelantov'a, and Ondv{r}ej Turek},
    journal={arXiv preprint arXiv:cs/0608065},
    year={2006},
    archivePrefix={arXiv},
    eprint={cs/0608065},
    primaryClass={cs.DM}
}