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 $S=\{n^2:1\le n\le \sqrt{N}\}$, in the sense that we have the additive energy estimate $E(A,S)\gg N\log N$. This is, in a sense, optimal.

Typos corrected and refined discussion

References in corpus (1)

• The inverse sieve problem in high dimensions