On linear versions of some addition theorems

TL;DR

This paper develops linear algebra analogues of classical additive number theory theorems, providing bounds on the dimensions of product subspaces in field extensions.

Contribution

It introduces linear versions of key additive theorems, offering new bounds on subspace dimensions in field extension contexts.

Findings

Lower bounds on dim(AB) in terms of dim(A) and dim(B)

Bounds on dim(B^n) based on initial subspace dimensions

Extension of additive number theory results to linear algebra setting

Abstract

Let K \subset L be a field extension. Given K-subspaces A,B of L, we study the subspace spanned by the product set AB = {ab | a \in A, b \in B}. We obtain some lower bounds on the dimension of this subspace and on dim B^n in terms of dim A, dim B and n. This is achieved by establishing linear versions of constructions and results in additive number theory mainly due to Kemperman and Olson.