# A new critical pair theorem applied to sum-free sets in Abelian groups

### Yahya ould Hamidoune

Université Pierre et Marie Curie, Paris, France### Alain Plagne

Ecole Polytechnique, Palaiseau, France

## Abstract

We shall prove a new generalization of Vosper critical pair theorem to finite Abelian groups. We next apply this new tool to the theory of $(k,l)$-free sets in finite Abelian groups. In particular, in most cases, we describe the structure of maximal $(k,l)$-free sets and determine the maximal cardinality of such a set. This result allows us for instance to give precisions on an old result of Yap: we are able to describe completely the maximal sum-free sets with cardinality at least one third of that of the ambient group.