An optimal L2 autoconvolution inequality

Let F denote the set of functions f : [-1/2, 1/2] -> R->= 0 such that. f = 1. We determine the value of inf f is an element of F vertical bar vertical bar f * f vertical bar vertical bar(2)(2) up to a 4 center dot 10(-6) error, thereby making progress on a problem asked by Ben Green. Furthermore, we prove that a unique minimizer exists. As a corollary, we obtain improvements on the maximum size of B-h[g] sets for (g, h) is an element of{(2, 2), (3, 2), (4, 2), (1, 3), (1, 4)}.