The supremum of autoconvolutions, with applications to additive number theory

2008/07/31 by Greg Martin, Kevin O'Bryant · 1 citation
Mathematics · #math.CA #math.CO #math.NT #msc:11B83 #msc:11P70 #msc:42A85

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published as Illinois Journal of Mathematics, Vol. 53 (2010), No. 1; 219-236. · 17 pages. to appear in IJM

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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