COMPRESSIONS, CONVEX GEOMETRY AND THE FREIMAN–BILU THEOREM

We 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 A ⊆ ℤ with small doubling. Our main result is the following. If ε > 0 and if A is a finite non-empty subset of a torsion-free abelian group with |A + A| ≤ K|A|, then A may be covered by e^KO(1) progressions of dimension ⌊ log 2 K + ε ⌋ and size at most |A|.