The popularity gap

TL;DR

This paper investigates the structure of difference sets in cyclic groups, establishing bounds on the number of representations of elements as differences and extending results to continuous, multidimensional, and dense subset cases.

Contribution

It proves a sharp bound on the second largest number of difference representations in cyclic groups and extends the results to continuous, multidimensional, and dense subset contexts.

Findings

The second largest difference representation count is approximately twice the average.

The bound of 2 is proven to be optimal.

Results are extended to continuous, multidimensional, and dense subset settings.

Abstract

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.