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 c | A | 3 solutions to a 1 + a 2 = a 3 + a 4 , a i ∈ A there exist sets A ′ ⊆ A and Λ ⊆ G , Λ = { λ 1 , … , λ t } , t ≪ c − 1 log | A | such that A ′ is contained in { ∑ j = 1 t ε j λ j | ε j ∈ { 0 , − 1 , 1 } } and A ′ has ≫ c | A | 3 solutions to a 1 ′ + a 2 ′ = a 3 ′ + a 4 ′ , a i ′ ∈ A ′ . We also study so-called symmetric sets or, in other words, sets of large values of convolution.