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B. Green, T. Tao, COMPRESSIONS, CONVEX GEOMETRY AND THE FREIMAN–BILU THEOREM, The Quarterly Journal of Mathematics, Volume 57, Issue 4, December 2006, Pages 495–504, https://doi.org/10.1093/qmath/hal009
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Abstract
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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