Some extensions of the Cauchy-Davenport theorem

The Cauchy-Davenport theorem states that, if p is prime and A, B are nonempty subsets of cardinality r, s   in Z/pZZ/pZ, the cardinality of the sumset A+B={a+b|a∈A,b∈B}A+B={a+b|a∈A,b∈B} is bounded below by min(r+s−1,p)min(r+s−1,p); moreover, this lower bound is sharp. Natural extensions of this result consist in determining, for each group G   and positive integers r,s⩽|G|r,s⩽|G|, the analogous sharp lower bound, namely the functionμG(r,s)=min{|A+B||A,B⊂G,|A|=r,|B|=s}. Important progress on this topic has been achieved in recent years, leading to the determination of μGμG for all abelian groups G. In this note we survey the history of earlier results and the current knowledge on this function.