2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/104747We note a link between combinatorial results of Bollobás 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|.9 pages, slight revisions in the light of comments from the referee. To appear in Quarterly Journal of Mathematics, OxfordNumber TheoryCombinatoricstext