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

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We study arithmetical and combinatorial properties of β-integers for β being the root of the equation x²=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 → Aᵐ⁻¹B and B→ Aᵐ⁻ⁿ⁻¹B. 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.