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; firstname.lastname@example.org; email@example.com; firstname.lastname@example.org
Accepted: 4 January 2007
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 uβ> coding distances between the consecutive β-integers, we determine precisely also the balance. The word uβ> is the only 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 the arithmetical properties of β-integers are related.
Mathematics Subject Classification: 68R15 / 11A63
Key words: Balance property / arithmetics / beta-expansions / infinite words
© EDP Sciences, 2007