Additive Correlation and the Inverse Problem for the Large Sieve

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