On Freiman's Theorem in a function field setting

We prove some new instances of a conjecture of Bachoc, Couvreur and Zémor 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.