Compressions, convex geometry and the Freiman-Bilu theorem

Ben Green; Terence Tao

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Compressions, convex geometry and the Freiman-Bilu theorem

Ben Green·Terence Tao

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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 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|.

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Ben Green, Terence Tao. 2006-03-03. Compressions, convex geometry and the Freiman-Bilu theorem. https://arxiv.org/abs/math/0511069

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