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(x) is in L1 and L2, 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 \|12 / I. This improves the previous bound of 0.591389 \| f \|12 / 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.
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