An almost-tight $L^2$ autoconvolution inequality

TL;DR

This paper precisely determines the minimal L^2 norm of the autoconvolution of functions with integral one, advancing a problem posed by Ben Green and deriving implications for bounds on certain combinatorial sets.

Contribution

It provides an almost exact value for the infimum of the L^2 norm of autoconvolutions and proves the uniqueness of the minimizer, addressing a longstanding open problem.

Findings

Exact value of the infimum up to 0.0014% error

Existence and uniqueness of the minimizer

Improved bounds on maximum size of specific B_h[g] sets

Abstract

Let $\mathcal{F}$ denote the set of functions $f \colon [-1/2,1/2] \to \mathbb{R}$ such that $\int f = 1$. We determine the value of $\inf_{f \in \mathcal{F}} \| f \ast f \|_2$ 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 $(g,h) \in \{ (2,2),(3,2),(4,2),(1,3),(1,4)\}$.