  We investigate the size of subspaces in sumsets and show two main results.
First, if A is a subset of F_2^n with density at least 1/2 - o(n^{-1/2}) then
A+A contains a subspace of co-dimension 1. Secondly, if A is a subset of F_2^n
with density at least 1/2-o(1) then A+A contains a subspace of co-dimension
o(n).
