  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.
