COMPRESSIONS, CONVEX GEOMETRY AND THE FREIMAN–BILU THEOREM

By Ben Joseph Green, Terence Tao. 2005

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We note a link between combinatorial results of Bollob\'as 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|.

Subfields: Applied Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/86a7d35e270cf6ed3255c87725be14e2534c49d0

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In Applied Mathematics

In Discrete Mathematics and Combinatorics

In Geometry and Topology

Citations in the corpus

The structure theory of set addition revisited (2012, background): “n exchange for hpKq. We just saw how to trade dimension in for size in the case where the group has bounded exponent. In general one cannot trade all of the dimension in for size but Green and Tao in [GT06] show (in torsion-free groups) how to reduce the dimension of the progression to OplogKq while”
The structure of approximate groups (2011, result): “roof. See [47] for the main part of this; the ﬁnal assertion is then a consequence of Ruzsa’s covering lemma, Lemma 5.1. For earlier results of this type with weaker bounds on rand P, see [9, 43]. In [25] it was noted that one can take ras small as ⌊log 2K+ ε⌋ for any ε>0, at the cost of decreasi”