Set addition in boxes and the Freiman-Bilu theorem

We show that if A is a large subset of a box in Z^d with dimensions L_1 >=
L_2 >= ... >= L_d which are all reasonably large, then |A + A| > 2^{d/48}|A|.
By combining this with Chang's quantitative version of Freiman's theorem, we
prove a structural result about sets with small sumset. If A is a set of
integers with |A + A| <= K|A|, then there is a progression P of dimension d <<
log K such that |A \cap P| >= \exp(-K^C)max (|A|, |P|). This is closely related
to a theorem of Freiman and Bilu, but is quantitatively stronger in certain
aspects.
[Added Oct 14th: I have temporarily withdrawn this paper, since Tao and I
have realised that a much stronger result follows by applying compressions (and
the Brunn-Minkowski theorem). This observation was inspired by a paper of
Bollobas and Leader which we were previously unaware of.
At some point soon this paper will be reinstated to prove just the result
that if A is a subset of R^d containing {0,1}^d then |A + A| >= 2^{d/48}|A|,
which may (possibly) be of independent interest.
Also soon, the joint paper of Tao and I should become available. ]