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 $(g,h) \in \{ (2,2),(3,2),(4,2),(1,3),(1,4)\}$.

中文翻译：

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最优自卷积不等式

令$\mathcal {F}$表示函数集$f \colon [-1/2,1/2] \to \mathbb {R}_{\geq 0}$使得$\int f = 1$。我们确定$\inf _{f \in \mathcal {F}} \|的值f \ast f \|_2^2$高达$4 \cdot 10^{-6}$错误，从而在 Ben Green 提出的问题上取得进展。此外，我们证明了唯一的最小化器的存在。作为推论，我们对$(g,h) \in \{ (2,2),(3,2),(4,2),(1 的$B_h[g]$集的最大大小进行了改进,3),(1,4)\}$。