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.

Based on joint works with Christine Bachoc, Alain Couvreur and Oriol Serra.

## Date:

Thu, 22/03/2018 - 15:30 to 16:30

## Location:

Manchester Building (Hall 2), Hebrew University Jerusalem