A polynomial bound in Freiman's theorem

In this paper the following improvement on Freiman's theorem on set addition is obtained (see Theorems 1 and 2 in Section 1). Let A subset of Z be a finite set such that \A + A\ < alpha \A\. Then A is contained in a proper d-dimensional progression P, where d less than or equal to [alpha - 1] and log(\P\/\A\) < Calpha(2)(logalpha)(3). Earlier bounds involved exponential dependence in alpha in the second estimate. Our argument combines I. Ruzsa's method, which we improve in several places, as well as Y. Bilu's proof of Freiman's theorem.