Sets with large additive energy and symmetric sets

TL;DR

This paper investigates the structure of sets with many additive solutions in finite Abelian groups, revealing that such sets contain large structured subsets with similar additive properties and exploring symmetric sets with high convolution values.

Contribution

It demonstrates that sets with many additive solutions contain large structured subsets within their span and analyzes symmetric sets with high convolution values.

Findings

Sets with many additive solutions contain large structured subsets.

Existence of subsets within A that are contained in the span of a small set L.

Analysis of symmetric sets with large convolution values.

Abstract

We show that for any set A in a finite Abelian group G that has at least c |A|^3 solutions to a_1 + a_2 = a_3 + a_4, where a_i belong A there exist sets A' in A and L in G, |L| \ll c^{-1} log |A| such that A' is contained in Span of L and A' has approximately c |A|^3 solutions to a'_1 + a'_2 = a'_3 + a'_4, where a'_i belong A'. We also study so-called symmetric sets or, in other words, sets of large values of convolution.