Let $(\mathcal{A}_i)_{i \in [s]}$ be a sequence of dense subsets of the Boolean cube $\{0,1\}^n$ and let $p$ be a prime. We show that if $s$ is assumed to be superpolynomial in $n$ then we can find distinct $i,j$ such that the two distributions of every mod-$p$ linear form on $\mathcal{A}_i$ and $\mathcal{A}_j$ are almost positively correlated. We also prove that if $s$ is merely assumed to be sufficiently large independently of $n$ then we may require the two distributions to have overlap bounded below by a positive quantity depending on $p$ only.

翻译：设$(\mathcal{A}_i)_{i \in [s]}$为布尔立方体$\{0,1\}^n$上一系列稠密子集，且$p$为素数。我们证明：若假设$s$在$n$上为超多项式增长，则可找到不同的$i,j$，使得$\mathcal{A}_i$和$\mathcal{A}_j$上每个模$p$线性形式的两个分布几乎正相关。此外，若仅假设$s$足够大且与$n$无关，则这两个分布的重叠度可被一个仅依赖于$p$的正数下界所约束。