  Suppose that A is a subset of the integers {1,...,N} of density a. We provide
a new proof of a result of Green which shows that A+A contains an arithmetic
progression of length exp(ca(log N)^{1/2}) for some absolute c>0. Furthermore
we improve the length of progression guaranteed in higher sumsets; for example
we show that A+A+A contains a progression of length roughly N^{ca} improving on
the previous best of N^{ca^{2+\epsilon}}.
