If B is a subset of G, we call an element g\in G "popular" if it can be written in many distinct ways as a difference b-b' with b, b' \in B. Popular difference sets appear naturally in the analytic approach to many problems in additive combinatorics, in the sense that many times the convolution of characteristic functions 1_C*1_D is used in order to make a statement about the difference set C-D. Very generally speaking, we will outline the limits of this approach. More precisely, we shall show that the set of popular differences of a large subset of G does not always contain the complete difference set of another large set. For this purpose we construct a so-called niveau set, which was first introduced by Ruzsa to show that there exists a large subset of Z/NZ whose sum set does not contain long arithmetic progressions.
Julia Wolf