On the expressive power of mod-$p$ linear forms on the Boolean cube

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.