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. The proofs combine probabilistic arguments with properties of Cayley sum graphs derived from their eigenvalues.