Compressions, convex geometry and the Freiman-Bilu theorem

BJ Green; TC Tao

doi:10.1093/qmath/hal009

Compressions, convex geometry and the Freiman-Bilu theorem

BJ Green, TC Tao

School of Mathematics

Research output: Contribution to journal › Article (Academic Journal) › peer-review

25 Citations (Scopus)

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
Original language	English
Pages (from-to)	495 - 504
Number of pages	10
Journal	Quarterly Journal of Mathematics
Volume	57 (4)
DOIs	https://doi.org/10.1093/qmath/hal009
Publication status	Published - Dec 2006

Bibliographical note

Publisher: Oxford University Press
Other: http://arxiv.org/abs/math/0511069

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10.1093/qmath/hal009

http://qjmath.oxfordjournals.org/cgi/content/abstract/57/4/495

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title = "Compressions, convex geometry and the Freiman-Bilu theorem",

abstract = "We note a link between combinatorial results of Bollob{\'a}s 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|.",

author = "BJ Green and TC Tao",

note = "Publisher: Oxford University Press Other: http://arxiv.org/abs/math/0511069",

year = "2006",

month = dec,

doi = "10.1093/qmath/hal009",

language = "English",

volume = "57 (4)",

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T1 - Compressions, convex geometry and the Freiman-Bilu theorem

AU - Green, BJ

AU - Tao, TC

N1 - Publisher: Oxford University Press Other: http://arxiv.org/abs/math/0511069

PY - 2006/12

Y1 - 2006/12

N2 - 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|.

AB - 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|.

U2 - 10.1093/qmath/hal009

DO - 10.1093/qmath/hal009

M3 - Article (Academic Journal)

VL - 57 (4)

SP - 495

EP - 504

JO - Quarterly Journal of Mathematics

JF - Quarterly Journal of Mathematics

SN - 0033-5606

ER -

Compressions, convex geometry and the Freiman-Bilu theorem

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