On Fourier coefficients of sets with small doubling

TL;DR

This paper demonstrates that subsets of finite abelian groups with small difference sets and small density have Fourier coefficients that imply correlation with large Bohr sets, with bounds close to optimal.

Contribution

It establishes a near-optimal relationship between small Fourier coefficients and correlation with large Bohr sets for sets with small doubling.

Findings

Fourier coefficients are small for sets with small doubling.

Such sets correlate with large Bohr sets.

Bounds on size and dimension are nearly tight.

Abstract

Let $A$ be a subset of a finite abelian group such that $A$ has a small difference set $A-A$ and the density of $A$ is small. We prove that, counter--intuitively, the smallness (in terms of $|A-A|$) of the Fourier coefficients of $A$ guarantees that $A$ is correlated with a large Bohr set. Our bounds on the size and the dimension of the resulting Bohr set are close to exact.