I will sketch a proof of the fact that for a prime p, every complement of a set of roughly \sqrt{p} elements of the finite field Z_p is a sumset, that is, is of the form A+A, whereas there are complements of sets of size roughly p^{2/3} which are not sumsets. This improves estimates of Green and Gowers, and can also be used to settle a recent problem of Nathanson.
Noga Alon