The popularity gap

Suppose that $A$ is a finite, nonempty subset of a cyclic group of either infinite or prime order. We show that if the difference set $A-A$ is ``not too large'', then there is a nonzero group element with at least as many as $(2+o(1))|A|^2/|A-A|$ representations as a difference of two elements of $A$; that is, the second largest number of representations is, essentially, twice the average. Here the coefficient $2$ is the best possible. We also prove continuous and multidimensional versions of this result, and obtain similar results for sufficiently dense subsets of an arbitrary abelian group.