# Additive Correlation and the Inverse Problem for the Large Sieve

## TL;DR

This paper investigates the additive structure of large integer sets avoiding certain residue classes, demonstrating a strong correlation with perfect squares, and establishing an optimal additive energy estimate.

## Contribution

It introduces a new inverse problem for the large sieve, showing that sets avoiding about half the residue classes modulo primes must have significant additive correlation with squares.

## Key findings

- Sets avoiding about p/2 residue classes modulo p have high additive energy with squares.
- The additive energy estimate E(A,S) is shown to be essentially optimal.
- The results connect sieve theory with additive combinatorics in a novel way.

## 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.