Higher moments of convolutions

TL;DR

This paper explores higher moments of convolutions of characteristic functions, generalizing additive energy, using combinatorics, Fourier analysis, and eigenvalues, with applications in additive combinatorics and number theory.

Contribution

It introduces a unified approach to higher energies of convolutions and demonstrates their applications in additive combinatorics.

Findings

Established basic properties of higher energies

Connected higher energies to classical additive energy concepts

Provided applications in additive combinatorics

Abstract

We study higher moments of convolutions of the characteristic function of a set, which generalize a classical notion of the additive energy. Such quantities appear in many problems of additive combinatorics as well as in number theory. In our investigation we use different approaches including basic combinatorics, Fourier analysis and eigenvalues method to establish basic properties of higher energies. We provide also a sequence of applications of higher energies additive combinatorics.