Towards a function field version of Freiman's Theorem

@article{Bachoc2017TowardsAF,
  title={Towards a function field version of Freiman's Theorem},
  author={Christine Bachoc and Alain Couvreur and Gilles Z'emor},
  journal={arXiv: Number Theory},
  year={2017}
}
We discuss a multiplicative counterpart of Freiman's $3k-4$ theorem in the context of a function field $F$ over an algebraically closed field $K$. Such a theorem would give a precise description of subspaces $S$, such that the space $S^2$ spanned by products of elements of $S$ satisfies $\dim S^2 \leq 3 \dim S-4$. We make a step in this direction by giving a complete characterisation of spaces $S$ such that $\dim S^2 = 2 \dim S$. We show that, up to multiplication by a constant field element… 

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