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

We study arithmetical and combinatorial properties of β-integers for β being the root of the equation x = mx − n,m, n ∈ N,m ≥ n + 2 ≥ 3. We determine with the accuracy of ±1 the maximal number of β-fractional positions, which may arise as a result of addition of two β-integers. For the infinite word uβ coding distances between consecutive β-integers, we determine precisely also the balance. The word uβ is the fixed point of the morphism A → AB and B → AB. In the case n = 1 the corresponding infinite word uβ is sturmian and therefore 1-balanced. On the simplest non-sturmian example with n ≥ 2, we illustrate how closely the balance and arithmetical properties of β-integers are related.