Subsets with a Small Sum II: the Critical Pair Problem

A basic problem raised by Kneser was to describe all the finite subsets A, B of an abelian group G such that | A+B | ≤ | A | + | B | − 1. Let G be an abelian group. The description of the pairsA , B⊂G such that | A+B | = | A | + | B | − 1 < |G | was considered in additive group theory. Vosper (1956) solved this problem completely for groups with a prime order. Kempermann’s theory for small sums describes the structure of these pairs, ifA+B is aperiodic or if there exists a uniquely expressible element inA+B. In this paper we study the same question with a fixed subset B satisfying the inequality: for all A such that 1 ≤ | A |