  Suppose that G is an abelian group, A is a finite subset of G with |A+A|<
K|A| and eta in (0,1] is a parameter. Our main result is that there is a set L
such that |A cap Span(L)| > K^{-O_eta(1)}|A| and |L| = O(K^eta log |A|). We
include an application of this result to a generalisation of the Roth-Meshulam
theorem due to Liu and Spencer.
