Set addition in boxes and the Freiman-Bilu theorem

Abstract

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 [Added Oct 14th: I have temporarily withdrawn this paper, since Tao and I have realised that a much stronger result follows by applying compressions (and the Brunn-Minkowski theorem).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. ]