On the product of vector spaces in a commutative field extension

Let $K \subset L$ be a commutative field extension. Given $K$-subspaces $A,B$ of $L$, we consider the subspace spanned by the product set $AB=\{ab \mid a \in A, b \in B\}$. If $\dim_K A = r$ and $\dim_K B = s$, how small can the dimension of be? In this paper we give a complete answer to this question in characteristic 0, and more generally for separable extensions. The optimal lower bound on $\dim_K$ turns out, in this case, to be provided by the numerical function $$\kappa_{K,L}(r,s) = \min_{h} (\lceil r/h\rceil + \lceil s/h\rceil -1)h,$$ where $h$ runs over the set of $K$-dimensions of all finite-dimensional intermediate fields $K \subset H \subset L$. This bound is closely related to one appearing in additive number theory.Comment: Submitted in November 200

Minimum product set sizes in nonabelian groups of order pq

Let G be a nonabelian group of order pq, where p and q are distinct odd primes. We analyze the minimum product set cardinality μG(r,s)=min|AB|μG(r,s)=min|AB|, where A and B range over all subsets of G of cardinalities r and s , respectively. In this paper, we completely determine μG(r,s)μG(r,s) in the case where G has order 3p and conjecture that this result can be extended to all nonabelian groups of order pq. We also prove that for every nonabelian group of order pq there exist 1⩽r,s⩽pq1⩽r,s⩽pq such that μG(r,s)>μZ/pqZ(r,s)μG(r,s)>μ[subscript Z over pqZ(r,s)].National Science Foundation (U.S.) (Grant DMS-0447070-001)United States. National Security Agency (Grant H98230-06-1-0013