Feasible schedules for rotating transmissions

Motivated by a scheduling problem that arises in the study of optical networks, we prove the following result, which is a variation of a conjecture of Haxell, Wilfong and Winkler. et k, n be two positive integers, let w _sj, 1 ≤ s ≤ n, 1 ≤ j ≤ k be nonnegative reals satisfying Σ_j=1^k w_sj < 1/n every 1 ≤ s ≤ n and let d_sj be arbitrary nonnegative reals. Then there are real numbers x₁, x₂,...,x_n such that for every j, 1 ≤j ≤ k, the n cyclic closed intervals I_s^(j) = [x_s + d_sj, x_s + d_dj + w _sj], (1 ≤ s ≤ n), where the endpoints are reduced modulo 1, are pairwise disjoint on the unit circle. The proof is based on some properties of multivariate polynomials and on the validity of the Dyson conjecture.