An almost-tight $L^2$ autoconvolution inequality

Let $\mathcal{F}$ denote the set of functions $f\text{\hspace{0.17em}⁣}:[-1\mathrm{/}2,1\mathrm{/}2]\to \mathbb{R}$ such that $\int f=1$. We determine the value of ${\inf }_{f\in \mathcal{F}}\mathrm{\parallel }f\ast f{\mathrm{\parallel }}_2$ 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 $(g,h)\in \{(2,2),(3,2),(4,2),(1,3),(1,4)\}$.