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.
Dividing lines in complete first-order theories were introduced by Shelah in the early seventies. A dividing line is a property such that the classes satisfying such a property have some nice behaviour while those not satisfying it have a bad one. Two of the best understood dividing lines are those of stability and superstability.
In this talk, I will study the notion of stability and superstability in abstract elementary classes of modules with respect to pure embeddings, i.e., classes of the form (K,≤p) where K is a class of R-modules for a fixed ring R and ≤p is the pure submodule relation. In particular, using that the class of p-groups with pure embeddings is a stable AEC, I will present a solution to Problem 5.1 in page 181 of Abelian Groups by László Fuchs. Moreover, I will show how the notion of superstability can be used to give new characterizations of noetherian rings, pure-semisimple rings, and perfect rings.
