The algebraicity of ill-distributed sets

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.

arXiv comment8 pages