Critical Pairs in Abelian Groups and Kemperman’s Theorem

A well-known result by Kemperman describes the structure of those pairs (A, B) of finite subsets of an abelian group satisfying |A+B| ≤ |A|+ |B| − 1. We establish a description which is, in a sense, dual to Kemperman’s, and as an application sharpen several results due to Deshouillers, Hamidoune, Hennecart, and Plagne. 1. Overview of the paper The sumset of two subsets A and B of an additively written group is denoted by A + B and defined as the set of all those group elements, representable as a sum of an element of A and an element of B: A +B := {a+ b : a ∈ A, b ∈ B}. In his remarkable paper [K60], Kemperman classified completely all pairs (A,B) of finite subsets of an abelian group with the small sumset; more precisely, those pairs satisfying |A+B| ≤ |A|+ |B| − 1. (1) This truly outstanding result, complementing a basic theorem by Kneser, may have not received yet the recognition that it certainly deserves. One of the reasons for this is that the structure of pairs, satisfying (1), is rather complicated, and so is Kemperman’s description reflecting this structure. Indeed, [K60] is not an easy reading, and an essential part of this paper constitutes an attempt to present Kemperman’s theorem and the mathematics around it in a possibly clear and transparent form, allowing one to appreciate this highly non-trivial theorem and facilitating its application. To prepare the ground and explain where Kemperman’s theorem stemmed from, we start with the theorems of Kneser and Kemperman-Scherk; this is the subject of the next section, where also the basic notion of period and some useful notation are introduced.