Some new results on higher energies

TL;DR

This paper introduces the method of higher energies to derive new upper bounds and structural insights for convex sets and sets with small product sets, advancing understanding in additive combinatorics.

Contribution

It develops the method of higher energies, providing new bounds and structural results for specific sets, and introduces dual popular difference sets.

Findings

New upper bounds for additive energies of convex sets

Structural results for higher sumsets

Introduction of dual popular difference sets

Abstract

In the paper we develop the method of higher energies. New upper bounds for the additive energies of convex sets, sets A with small |AA| and |A(A+1)| are obtained. We prove new structural results, including higher sumsets, and develop the notion of dual popular difference sets.