On Freiman's Theorem in a function field setting

TL;DR

This paper extends Freiman's Theorem to a function field context, proving new cases of a conjecture that characterizes the structure of certain finite-dimensional vector spaces over rational function fields.

Contribution

It provides new instances of a conjecture generalizing Freiman's 3k-4 Theorem to multiplicative settings in function fields, with specific conditions on vector space dimensions.

Findings

Confirmed the conjecture for rational function fields over algebraically closed fields when S^2 = 2 S + 1

Identified structural properties of vector spaces satisfying the dimension condition

Extended Freiman's Theorem to a broader algebraic setting

Abstract

We prove some new instances of a conjecture of Bachoc, Couvreur and Z\'emor that generalizes Freiman's $3k-4$ Theorem to a multiplicative version in a function field setting. As a consequence we find that if $F$ is a rational function field over an algebraically closed field $K$ and $S \subset F$ a finite dimensional $K$-vector space such that $\dim S^2 = 2\dim S + 1$, then the conjecture holds.