  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.
