  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.
