On classical inequalities for autocorrelations and autoconvolutions

TL;DR

This paper investigates an autocorrelation inequality related to additive combinatorics, establishing extremizers, approximating optimal constants numerically, and exploring related autoconvolution problems.

Contribution

It proves the existence of extremizers for a broad class of weights and provides numerical approximations for the best constants in the inequality.

Findings

Existence of extremizers for general weights including Gaussian and characteristic functions

Numerical approximation of the optimal constants in the inequality

Discussion of related autoconvolution problems

Abstract

In this paper we study an autocorrelation inequality proposed by Barnard and Steinerberger. The study of these problems is motivated by a classical problem in additive combinatorics. We establish the existence of extremizers to this inequality, for a general class of weights, including Gaussian functions (as studied by the second author and Ramos) and characteristic function (as originally studied by Barnard and Steinerberger). Moreover, via a discretization argument and numerical analysis, we find some almost optimal approximation for the best constant allowed in this inequality. We also discuss some other related problem about autoconvolutions.