Compressions, convex geometry and the Freiman-Bilu theorem
Number Theory
2007-05-23 v2 Combinatorics
Abstract
We note a link between combinatorial results of Bollob\'as and Leader concerning sumsets in the grid, the Brunn-Minkowski theorem and a result of Freiman and Bilu concerning the structure of sets of integers with small doubling. Our main result is the following. If eps > 0 and if A is a finite nonempty subset of a torsion-free abelian group with |A + A| <= K|A|, then A may be covered by exp(K^C) progressions of dimension [log_2 K + eps] and size at most |A|.
Cite
@article{arxiv.math/0511069,
title = {Compressions, convex geometry and the Freiman-Bilu theorem},
author = {Ben Green and Terence Tao},
journal= {arXiv preprint arXiv:math/0511069},
year = {2007}
}
Comments
9 pages, slight revisions in the light of comments from the referee. To appear in Quarterly Journal of Mathematics, Oxford