Additive combinatorics enable one to characterize subsets S of elements in a group such that S+S has small cardinality. We are interested in linear analogues of these results, namely characterizing subspaces S in some algebras (mostly extension fields) such that the linear span of the set S^2 of products st, for s,t in S, has small dimension. We shall present a linear analogue of a theorem of Vosper which says that under the right conditions, a sufficiently small dimension for S^2 implies that S has a basis of elements in geometric progression. We shall also attempt to go beyond this theorem and claim that when the dimension of S^2 is sufficiently small, S must be close to a Riemann-Roch space of an algebraic curve of small genus.
Based on joint works with Christine Bachoc, Alain Couvreur and Oriol Serra.