We improve an estimate for the additive energy of sets \(A\) with small product \(AA\). The proof uses some properties of level sets of convolutions of the indicator function of \(A\), namely, their almost invariance under multiplication by elements of \(A\).

中文翻译：

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小乘性加倍集的流行和与差

我们用小乘积\（AA \）改进了集合\（A \）的附加能量的估计。该证明使用了\（A \）指标函数的卷积水平集的某些属性，即它们在与\（A \）元素相乘的情况下几乎不变。