A linear Analogue of Kneser's Theorem and Related Problems

A theorem of Kneser in additive combinatorics states that in an abelian group $G$ if $A$ and $B$ are finite subsets in $G$ and $AB=\{ab\mid a\in A, b\in B\}$ then $|AB|\geq |A|+|B|-|H(AB)|$, where $H(AB)=\{g\mid g\in G, g(AB)=AB\}$. More than fifteen years ago, motivated by the study of a problem about finite fields, we (jointly with Xiang-Dong Hou and Ka Hin Leung) proved an analogous result for vector spaces over a field $E$ in an extension field $K$ of $E$, which is now called a linear analogue of Kneser's theorem. This linear analogue has found some interesting applications and motivated further investigations. We will talk about this linear analogue of Kneser's theorem and related problems.