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

Doppler Institute for Mathematical Physics and Applied Mathematics and Department of Mathematics, FNSPE, Czech Technical University, Trojanova 13, 120 00 Praha 2, Czech Republic; l.balkova@centrum.cz; masakova@km1.fjfi.cvut.cz; oturek@centrum.cz

Received: 29 August 2006
Accepted: 4 January 2007

Abstract

We study some arithmetical and combinatorial properties of β-integers for β being the larger root of the equation x² = mx - n,m,n ∈ ℵ, 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 u_β> coding distances between the consecutive β-integers, we determine precisely also the balance. The word u_β> is the only fixed point of the morphism A → A^m-1B and B → A^m-n-1B. In the case n = 1, the corresponding infinite word u_β> is sturmian, and, therefore, 1-balanced. On the simplest non-sturmian example with n ≥ 2, we illustrate how closely the balance and the arithmetical properties of β-integers are related.