Suppose that A is a finite, non-empty 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 non-zero 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.
This video was produced by the International Centre for Mathematical Sciences, as part of the workshop UK-Vietnam mathematics joint meeting.
