An Isoperimetric Method in Additive Theory

The main result has the following corollary. LetGbe a group containing two finite subsetsAandBsuch that every element ofG\{1} has order ≥max(|A|, |B|). Suppose 2≤min(|A|, |B|) and |AB|=|A|+|B|−1<|G|−1. Assume that 1∈Band thatGis generated byB. Then there arex,y,r∈Gsuch thatA={xri|0≤i≤|A|−1} andB={riy|0≤i≤|B|1}. Applying this result with |G| prime, one obtains a well-known theorem of Vosper. The case whereGis torsion free is a recent result of Brailovski and Freiman.