A Half-Constant Upper Bound for Chowla's Cosine Problem

Source abstract

Let lambda(n) denote the infimum of -min_x sum_{a in A} cos(ax) over all sets A of n positive integers. While the classical Sidon-difference construction yields lambda(n) infinity} lambda(n)/sqrt(n) = cn^(1/4) for every n-element set A subset of [1, Cn]; and (2) (-min f_A)^3 ||f_A||_1^2 >= cn for every n-element set A subset of N. The second bound recovers the 1/5 exponent of Bedert and Shankar and improves it whenever ||f_A||_1 is small relative to -min f_A. Numerical values of the construction are reported up to |A| = 25,440.