Pl\"unnecke and Kneser type theorems for dimension estimates

TL;DR

This paper extends classical additive combinatorics theorems like Pl"unnecke and Kneser to the setting of division rings and associative algebras, providing new dimension estimates for Minkowski products.

Contribution

It introduces analogues of Pl"unnecke and Kneser theorems for dimension estimates in division rings and associative algebras, and generalizes Tao's small doubling classification to division rings.

Findings

Derived dimension bounds for Minkowski products in division rings.

Extended Pl"unnecke and Kneser theorems to associative algebras.

Classified small doubling sets in division rings similar to Tao's results.

Abstract

Given a division ring K containing the field k in its center and A,B two finite subsets of K\{0}, we give some analogues of Pl\"unnecke and Kneser theorems for the dimension of the k-linear span of the Minkowski product AB in terms of the dimensions of the k-linear spans of A and B. These Pl\"unnecke type estimates are then generalized to the case of associative algebras. We also obtain an analogue in the context of division rings of a theorem by Tao classifying the sets of small doubling in a group.