Energies and Structure of Additive Sets

In this paper we prove that any sumset or difference set has large E 3 energy. Also, we give a full description of families of sets having critical relations between some kind of energies such as E k , T k and Gowers norms. In particular, we give criteria for a set to be a • set of the form H Λ, where H + H is small and Λ has " random structure " , • set equal to a disjoint union of sets H j each with small doubling, • set having a large subset A with 2A equal to a set with small doubling and |A + A | ≈ |A| 4 /E(A).