## Section: New Results

### Towards a function field version of Freiman's theorem

In a collaboration with Christine Bachoc and Gilles Zémor (University of Bordeaux), A. Couvreur obtained a characterisation of subspaces $S$ of a function field $F$ over an algebraically closed field satisfiying

where ${S}^{2}$ denotes the space spanned by all the products of two elements of $S$. They obtained the following result [18]:

**Theorem.**
*Let $F$ be a function field over an algebraically closed field,
and $S$ be a finite dimensional subspace of $F$ which
spans $F$ as an algebra and such that*

*Then $F$ is a function field of transcendence degree 1 and*