On small sumsets in abelian groups

In this paper we investigate the structure of those pairs of finite subsets of an abelian group whose sums have relatively few elements: \A + B\ < \A\ + \B\. In 1960, J. H. B. Kemperman gave an exhaustive but rather sophisticated description of recursive nature. Using intermediate results of Kemperman, we obtain below a description of another type. Though not (generally speaking) sufficient, our description is intuitive and transparent and can be easily used in applications.