The inverse sieve problem in high dimensions

TL;DR

This paper proves that large sets of integer points in high dimensions, which occupy few residue classes modulo many primes, must essentially be solutions to low-degree polynomial equations, answering a question by Helfgott and Venkatesh.

Contribution

It establishes a high-dimensional inverse sieve theorem linking residue class distribution to algebraic structure, extending previous one-dimensional results.

Findings

Sets with few residue classes mod p are contained in low-degree polynomial solutions.

The result applies to high-dimensional integer points in [0,N]^d.

Answers an open question by Helfgott and Venkatesh.

Abstract

We show that if a big set of integer points in [0,N]^d, d>1, occupies few residue classes mod p for many primes p, then it must essentially lie in the solution set of some polynomial equation of low degree. This answers a question of Helfgott and Venkatesh.