Combinatorial and arithmetical properties of infinite words associated with non-simple quadratic parry numbers
We study some arithmetical and combinatorial properties of beta-integers for beta being the larger root of the equation x(2) = mx - n, m, n epsilon N, m >= n + 2 >= 3. We determine with the accuracy of +/- 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 the consecutive beta- integers, we determine precisely also the balance. The word u(beta) is the only fixed point of the morphism A -> A (m-1) B and B -> 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 = 2, we illustrate how closely the balance and the arithmetical properties of beta- integers are related.
arithmetics beta balance property expansions infnite words