Chowla's cosine problem

TL;DR

This paper investigates the structure of finite symmetric subsets in discrete abelian groups, establishing conditions under which they resemble subgroups or exhibit large Fourier coefficients, advancing understanding of harmonic analysis in algebraic structures.

Contribution

It provides new results characterizing when symmetric subsets are close to subgroups or have significant Fourier coefficients in discrete abelian groups.

Findings

Either the set is close to a union of few subgroups or has a large Fourier coefficient.

In finite groups, large symmetric sets are either near a subgroup or have a large Fourier transform.

The results quantify the structure of symmetric subsets in terms of subgroup approximation or Fourier analysis.

Abstract

Suppose that G is a discrete abelian group and A is a finite symmetric subset of G. We show two main results. i) Either there is a set H of O(log^c|A|) subgroups of G with |A \triangle \bigcup H| = o(|A|), or there is a character X on G such that -wh{1_A}(X) >> log^c|A|. ii) If G is finite and |A|>> |G| then either there is a subgroup H of G such that |A \triangle H| = o(|A|), or there is a character X on G such that -wh{1_A}(X)>> |A|^c.