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 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. [ABSTRACT FROM AUTHOR]