The supremum of autoconvolutions, with applications to additive number theory

Abstract

We adapt a number-theoretic technique of Yu to prove a purely analytic theorem: if $f\in L^1({\mathbb R}) \cap L^2({\mathbb R})$ is nonnegative and supported on an interval of length $I$, then the supremum of $f\ast f$ is at least $0.631 \|f\|_1^2/I$. This improves the previous bound of $0.591389 \|f\|_1^2/I$. Consequently, we improve the known bounds on several related number-theoretic problems. For a set $A\subseteq\{1,2,\dots,n\}$, let $g$ be the maximum multiplicity of any element of the multiset $\{a_1+a_2 : a_i\in A\}$. Our main corollary is the inequality $g n > 0.631 |A|^2$, which holds uniformly for all $g$, $n$, and $A$.