Some extensions of the Cauchy-Davenport theorem

Abstract

Abstract The Cauchy-Davenport theorem states that, if p is prime and A, B are nonempty subsets of cardinality r, s in Z / p Z , the cardinality of the sumset A + B = { a + b | a ∈ A , b ∈ B } is bounded below by 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 | , the analogous sharp lower bound, namely the function μ G ( 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 μ G for all abelian groups G. In this note we survey the history of earlier results and the current knowledge on this function.