math.CA — Classical Analysis and ODEs

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

Contributed by Tejasvi Singh Tomar

Let $C_1$ be the largest constant such that $\sup_{|t|\le 1/2}\int f(t-x)f(x)\,dx\ge C_1\bigl(\int_{-1/4}^{1/4}f\bigr)^2$ for every nonnegative function $f$ on the real line. We prove $C_1\ge 1.337$, 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 $\{1,\dots,N\}$ has fewer than $1.2231\sqrt{gN}$ elements once $N$ is large enough in terms of $g$. For real-valued $f$ supported in $[-1/4,1/4]$, let $C_3$ and $C_3'$ 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 $C_3\ge C_3'\ge 4/3$ with the argument principle. Both proofs are computer-assisted, and both theorems are also proved in Lean 4.

Author comments: Published on Zenodo: https://doi.org/10.5281/zenodo.23715655