Sets with large additive energy and symmetric sets

We show that for any set A in a finite Abelian group G that has at least c3|A| 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 and A′ has ≫c3|A| solutions to , . We also study so-called symmetric sets or, in other words, sets of large values of convolution.