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COMPRESSIONS, CONVEX GEOMETRY AND THE FREIMAN-BILU THEOREM

Authors:	Green B; Tao T

Institution:	¹ Department of Mathematics, University of Bristol, University Walk, Bristol BS8 1TW, UK ² Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA

Abstract:	We note a link between combinatorial results of Bollobásand Leader concerning sumsets in the grid, the Brunn–Minkowskitheorem and a result of Freiman and Bilu concerning the structureof sets A ${subseteq}$ $Z$ with small doubling. Our main result is the following. If ${varepsilon}$ > 0 and if A is a finitenon-empty subset of a torsion-free abelian group with \|A + A\| $≤$ K\|A\|, then A may be covered by e^K^O(1) progressions of dimension ${lfloor}$ log ₂ K + ${varepsilon}$ ${rfloor}$ and size at most \|A\|.

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