Sets of integers with large trigonometric sums

We investigate the problem of optimizing, for a fixed integer k and real u, and on all sets K = {a(1) < a(2) < ... < a(k)} subset of Z, the measure of the set of alpha is an element of [0, 1] where the absolute Value of the trigonometric sum S-K (alpha) = Sigma(j=1)(k) e(2 pi i alpha aj) is greater than Ic - u. When u is sufficiently small with respect to k we are able to construct a set K-ex which is very close to optimal. This set is a union of a finite number of arithmetic progressions. We are able to show that any more optimal set, if one exists, has a similar structure to that of K-ex. We also get tight upper and lower bounds on the maximal measure.