Certified lower bounds for the first and the signed third autocorrelation constants

Let C₁ be the largest constant such that sup_{|t| ≤ 1/2} ∫ f(t−x) f(x) dx ≥ C₁ (∫_−1/4^1/4 f)² for every nonnegative function f on the real line. We prove C₁ ≥ 1.337, improving the bound 1.2937 of Kim and Pilanci, by a spatial branch-and-bound over the masses of f on equal bins. Consequently, for every large g, every g-Sidon set in {1, …, N} has fewer than 1.2231√(gN) elements once N is large enough in terms of g. For real-valued f supported in [−1/4, 1/4], let C₃ and C₃′ be the analogous constants for the essential suprema of |f∗f| and of f∗f; no lower bound better than the trivial bound 1 was recorded for them. We prove C₃ ≥ C₃′ ≥ 4/3 with the argument principle. Both proofs are computer-assisted, and both theorems are also proved in Lean 4.

The programs, the Lean formalization and the certificates, except the logs of the search for C₁, which are too large for this archive, are archived at doi:10.5281/zenodo.23715661 and available at github.com/tejasvi/autocorrelation-lower-bounds/tree/v13 (commit bfabb085b8c4).