Subsets with Small Sums in Abelian Groups' I: the Vosper Property

LetGbe an abelian group containing a finite subsetBsuch that, for every non-empty finite subsetA⊂G, |A+B|≥min(|G|,|A|+|B|-1).We obtain the necessary and sufficient condition for the validity of the stronger property:For every finite subset A⊂G, such that |A|≥2, |A+B|≥min(|G|-1,|A|+|B|).We apply our methods to the range of diagonal forms over finite fields, obtaining a new proof of a result of Tietäväinen. Our proof works in characteristic 2, where the question was open. We also apply our methods to obtain a new characterization for abelian Cayley graphs for which each minimum cutset originates or ends in a vertex.