The supremum of autoconvolutions, with applications to additive number theory

Details The supremum of autoconvolutions, with applications to additive number theory The supremum of autoconvolutions, with applications to additive number theory

2008-07-31

arxiv.org/abs/0807.5121v2

Illinois Journal of Mathematics, Vol. 53 (2010), No. 1; 219-236.

Mathematics

Classical Analysis and ODEs

17 pages. to appear in IJM

Scientific paper

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.

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