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 {subseteq} Z with small doubling.
Our main result is the following. If {varepsilon} > 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 {lfloor} log 2 K + {varepsilon} {rfloor} and size at most |A|.
Translated title of the contribution | Compressions, convex geometry and the Freiman-Bilu theorem |
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Original language | English |
Pages (from-to) | 495 - 504 |
Number of pages | 10 |
Journal | Quarterly Journal of Mathematics |
Volume | 57 (4) |
DOIs | |
Publication status | Published - Dec 2006 |
Bibliographical note
Publisher: Oxford University PressOther: http://arxiv.org/abs/math/0511069