Sets with large additive energy and symmetric sets

AbstractWe show that for any set A in a finite Abelian group G that has at least c|A|3 solutions to a1+a2=a3+a4, ai∈A there exist sets A′⊆A and Λ⊆G, Λ={λ1,…,λt}, t≪c−1log|A| such that A′ is contained in {∑j=1tεjλj|εj∈{0,−1,1}} and A′ has ≫c|A|3 solutions to a1′+a2′=a3′+a4′, ai′∈A′. We also study so-called symmetric sets or, in other words, sets of large values of convolution