Bounds for the sign uncertainty constant in dimension one

Bourgain, Clozel and Kahane proved that a nonzero even real function f on the real line with f and f̂ integrable, f(0) ≤ 0 and f̂(0) ≤ 0 cannot have f and f̂ both nonnegative outside an arbitrarily small neighborhood of the origin. Let A(f) be the radius beyond which f is nonnegative, and let A+(1)2 be the largest constant such that A(f)A(f̂) ≥ A+(1)2 for all such f. This is the constant of Problem 6.11 in the list of Georgiev, Gómez-Serrano, Tao and Wagner, where f(0) < 0 and f̂(0) < 0 are required. The previous bounds are 0.2025 ≤ A+(1)2 ≤ 0.3089, due to Gonçalves, Oliveira e Silva and Steinerberger and to a computer search reported by Chung, Du and Wesley. We prove 0.5502 < A+(1) ≤ 0.5509, so that A+(1)2 lies in (0.30272004, 0.30349081]. For the lower bound, since extremizers exist, it suffices to show that every even real integrable g with ĝ = g, g(0) ≤ 0 and g ≥ 0 on |x| > r vanishes. The proof uses a weight that comes from explicit differences between Gaussians and their Fourier transforms and from cosines, corrected on [−r, r] by an approximate solution of a Fredholm integral equation; an a priori bound absorbs the residual of this approximation. Two separately written programs verify the required inequalities for r = 0.5502 in ball arithmetic. The upper bound comes from an explicit self-dual combination of Gaussians whose positivity beyond 0.5509 is verified in the same way. The certificates and the programs are archived at doi:10.5281/zenodo.23509816 and available at github.com/tejasvi/sign-uncertainty-constant.