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 eKO(1) progressions of dimension ⌊ log 2 K + ε ⌋ and size at most |A|.

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