A Generalization of an Addition Theorem of Kneser Original Research Article

A theorem of Kneser states that in an abelian group G, if A and B are finite subsets in G and AB={ab:aset membership, variantA,bset membership, variantB}, then midABmidgreater-or-equal, slantedmidAmid+ midBmid− midH(AB)mid where H(AB)={gset membership, variantG:g(AB)=AB}. Motivated by the study of a problem in finite fields, we prove an analogous result for vector spaces over a field E in an extension field K of E. Our proof is algebraic and it gives an immediate proof of Kneserʹs Theorem.