The structure of popular difference sets

We show that the set of popular differences of a large subset of ℤ _N 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 in Ruz87] and later used in Ruz91] to show that there exists a large subset of ℤ _N whose sumset does not contain any long arithmetic progressions. In this paper we make substantial use of measure concentration results on the multi-dimensional torus and Esseen’s Inequality.