Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4652885 | Electronic Notes in Discrete Mathematics | 2007 | 8 Pages |
Abstract
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.
Keywords
Related Topics
Physical Sciences and Engineering
Mathematics
Discrete Mathematics and Combinatorics
Authors
Shalom Eliahou, Michel Kervaire,