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

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) <= (1 + o(1))sqrt(n)/2, we construct sets achieving a leading constant of 1/2: lambda(n) <= (1/2)sqrt(n) + O(n^(1/4)sqrt(log n)), establishing limsup_{n->infinity} lambda(n)/sqrt(n) <= 1/2. This construction combines a two-sided Sidon square identity, a Singer difference set, a Fejer-kernel majorant, a parity-biased deletion, and an exact-cardinality probabilistic selection. We also provide an L^4 criterion and a multiplicity bound constraining any construction with a smaller leading constant. In the opposite direction, we incorporate the L^1 norm into Bedert's boundary estimate with an explicit constant, yielding two lower bounds for f_A(x) = sum_{a in A} cos(ax): (1) -min f_A >= 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.