  We show two results. First, a refinement of Freiman's theorem: if A is a
finite set of integers and |A+A| < K|A|, then A is contained in a
multidimensional progression of dimension at most O(K^{7/4} log^3K) and size at
most exp(O(K^{7/4} log^3K))|A|. Secondly, an improvement of a result of
Konyagin and Laba: if A is a finite set of reals and a is a transcendental then
|A+aA| >> |A|(log |A|)^{4/3-\epsilon} for all \epsilon>0.
