Bounds for the sign uncertainty constant in dimension one

Bourgain, Clozel and Kahane proved that a nonzero even real function f on ℝ 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. The sharp constant in this sign uncertainty principle is A₊(1), the infimum of √(A(f)A(f̂)) over all such f, where A(f) is the radius beyond which f is nonnegative; its square is the constant of Problem 6.11 in the list of Georgiev, Gómez-Serrano, Tao and Wagner. We prove 0.5502 < A₊(1) ≤ 0.5509, improving the previous bounds 0.45 ≤ A₊(1) ≤ √0.3089; in particular A₊(1) < 0.555, as conjectured by Gonçalves, Oliveira e Silva and Ramos. The lower bound comes from a weight in the spirit of the summation formulas of Cohn and Gonçalves, built from Gaussians and cosines and corrected near the origin by an approximate solution of a Fredholm integral equation, and the upper bound from an explicit self-dual combination of Gaussians. The numerical inequalities in both proofs are verified in ball arithmetic, and the bounds 0.5502 ≤ A₊(1) ≤ 0.5509 also have formal proofs in Lean 4.

The certificates, the programs and the Lean proofs are archived at doi:10.5281/zenodo.23716093 and available at github.com/tejasvi/sign-uncertainty-constant/tree/v6 (commit 93c4969f2305).