Hyper-atoms and the critical pair Theory

We introduce the notion of a hyper-atom. One of the main results of this paper is the 2|G| 3 –Theorem: Let S be a finite generating subset of an abelian group G of order ≥ 2. Let T be a finite subset of G such that 2 ≤ |S| ≤ |T |, S + T is aperiodic, 0 ∈ S ∩ T and 2|G|+ 2 3 ≥ |S + T | = |S|+ |T | − 1. Let H be a hyper-atom of S. Then S and T are H–quasi-periodic. Moreover φ(S) and φ(T ) are arithmetic progressions with the same difference, where φ : G 7→ G/H denotes the canonical morphism. This result implies easily the traditional critical pair Theory and its basic stone: Kemperman’s Structure Theorem.