Sets of integers with large trigonometric sums Academic Article uri icon

abstract

  • We investigate the problem of optimizing, for a fixed integer k and real u and on all sets K = {a1 < a2 < ⋯ < ak} ⊂ ℤ, the measure of the set of α ∈ [0, 1] where the absolute value of the trigonometric sum SK(α) = ∑j=1k e2πiαaj is greater than k - u. When u is sufficiently small with respect to k we are able to construct a set Kex 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 Kex. We also get tight upper and lower bounds on the maximal measure.

publication date

  • January 1, 1999