On the union of arithmetic progressions

We show that for any integer n ≥ 1 and real ε > 0, the union of n arithmetic progressions with pairwise distinct differences, each of length n, contains at least c (ε)n^2-ε elements, where c (ε) is a positive constant depending only on ε. This estimate is sharp in the sense that the assertion becomes invalid for ε = 0. We also obtain estimates for the "asymmetric case" where the number of progressions is distinct from their lengths.