The supremum of autoconvolutions, with applications to additive number theory

TL;DR

This paper proves a new lower bound for the supremum of autoconvolutions of functions supported on an interval, leading to improved bounds in additive number theory problems.

Contribution

It introduces an analytic approach to bound autoconvolutions, enhancing previous results and applying these to derive stronger bounds in additive number theory.

Findings

Improved lower bound for autoconvolution supremum: 0.631 

Enhanced bounds on subset sum multiplicities in additive number theory

Application of analytic techniques to number-theoretic problems

Abstract

We adapt a number-theoretic technique of Yu to prove a purely analytic theorem: if f(x) is in L^1 and L^2, is nonnegative, and is supported on an interval of length I, then the supremum of the convolution f*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 subset A of {1,2, ..., n}, let g be the maximum multiplicity of any element of the multiset {a+b: a,b in A}. Our main corollary is the inequality gn>0.631|A|^2, which holds uniformly for all g, n, and A.