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$L^2$ autoconvolution inequality
Let 
$\mathcal {F}$ denote the set of functions 
$f \colon [-1/2,1/2] \to \mathbb {R}_{\geq 0}$ such that 
$\int f = 1$. We determine the value of 
$\inf _{f \in \mathcal {F}} \| f \ast f \|_2^2$ up to a 
$4 \cdot 10^{-6}$ 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 
$(g,h) \in \{ (2,2),(3,2),(4,2),(1,3),(1,4)\}$.
