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 = {a₁ < a₂ < ⋯ < a_k} ⊂ ℤ, the measure of the set of α ∈ [0, 1] where the absolute value of the trigonometric sum S_K(α) = ∑_j=1^k e^2πiαaj is greater than k - 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.