Let A ⊆ [1, N] be a set of integers with |A| ≫ $\sqrt N$. We show that if A avoids about p/2 residue classes modulo p for each prime p, then A must correlate additively with the squares S = {n2 : 1 ≤ n ≤ $\sqrt N$}, in the sense that we have the additive energy estimate $$ E(A,S)\gg N\log N. $$ This is, in a sense, optimal.

中文翻译：

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大筛子的加性相关和反问题

让一种⊆ [1,ñ] 是一组具有 | 的整数一种| ≫$\sqrt N$. 我们证明如果一种避免约p/2 残基类模p对于每个素数p， 然后一种必须与平方相加相关小号= {n2: 1 ≤n≤$\sqrt N$}，从某种意义上说，我们有加性能量估计$$ E(A,S)\gg N\log N. $$从某种意义上说，这是最优的。