An almost-tight $L^2$ autoconvolution inequality

Abstract

Let $\mathcal{F}$ denote the set of functions $f:[-1/2,1/2]\to \mathbb{R}$ such that $\int f=1$. We determine the value of ${\inf }_{f\in \mathcal{F}}\Vert f\ast f{\Vert }_2$ up to a 0.0014\% 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 $(g,h)\in \{(2,2),(3,2),(4,2),(1,3),(1,4)\}$.

18 pages, 1 figure, 1 table