Energies and Structure of Additive Sets

Abstract

In this paper we prove that any sumset or difference set has large $\textsf{E}_3$ energy. Also, we give a full description of families of sets having critical relations between some kind of energies such as $\textsf{E}_k$, $\textsf{T}_k$ and Gowers norms. In particular, we give criteria for a set to be a 

• set of the form $H\dotplus \Lambda$, where $H+H$ is small and $\Lambda$ 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 $|A'+A'| \approx |A|^4 / \textsf{E}(A)$.