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 | < ∞, |A + B | ≥ min(| G |, | A | + |B | − 1). We obtain a recursive description for the subsets A such that |A + B | ≤ | A | + | B | − 1. As corollary of our description, we obtain the following result which implies some limitations of Kempermann’s theory. Suppose that B is neither a coprogression nor almost periodic and that 2 ≤ |A | ≤ | G | − | B | − 1. If |A + B | = | A | + | B | − 1, then A is periodic and A + B contains no unique expression elements. The results obtained in this section are strongly based on those obtained in Part I.