The algebraicity of ill-distributed sets

TL;DR

This paper proves that sets with limited residue class distribution in multi-dimensional integers are structurally close to solutions of low-degree polynomial equations, revealing a new fundamental property of such sets.

Contribution

It establishes the first structural theorem for arbitrary t < d, showing these sets lie near polynomial solution sets with degree bounded by a polylogarithm.

Findings

Sets occupy fewer than p^t residue classes for all primes p.

Such sets are essentially contained in low-degree polynomial solution sets.

Degree of these polynomials is at most polylogarithmic in N.

Abstract

We show that every set S in [N]^d occupying less than p^t residue classes for some real number t < d and every prime p, must essentially lie in the solution set of a polynomial equation of degree at most (log N)^C, for some constant C depending only on t and d. This provides the first structural result for arbitrary t < d and S.