Energies and structure of additive sets

TL;DR

This paper investigates the energies and structural properties of additive sets, providing new insights into their sumsets, difference sets, and criteria for specific additive configurations.

Contribution

It establishes that sumsets and difference sets have large E_3 energy and characterizes sets with critical energy relations, including criteria for structured and random-like sets.

Findings

Sumsets and difference sets have large E_3 energy.

Characterization of sets with specific energy relations and structures.

Criteria for sets being of the form H+L, unions of small doubling sets, or having large structured subsets.

Abstract

In the 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 1) set of the form H+L, where H+H is small and L has "random structure", 2) set equals a disjoint union of sets H_j, each H_j has small doubling, 3) set having large subset A' with 2A' is equal to a set with small doubling and |A'+A'| \approx |A|^4 / \E(A).