A Szemeredi-type regularity lemma in abelian groups, with applications

BJ Green

Research output: Contribution to journalArticle (Academic Journal)peer-review

115 Citations (Scopus)

Abstract

Szemeredi's regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemeredi's regularity lemma in the context of abelian groups and use it to derive some results in additive number theory. One is a structure theorem for sets which are almost sum-free. If A subset of {1,..., N} has delta N-2 triples (a(1), a(2), a(3)) for which a(1) + a(2) = a(3) then A = B boolean OR C, where B is sum-free and | C| = delta'N, and delta' --> 0 as delta --> 0. Another answers a question of Bergelson, Host and Kra. If alpha, epsilon > 0, if N > N-0(alpha, epsilon) and if A subset of {1,..., N} has size alpha N, then there is some d not equal 0 such that A contains at least (alpha(3)-epsilon) N three-term arithmetic progressions with common difference d.
Translated title of the contributionA Szemeredi-type regularity lemma in abelian groups, with applications
Original languageEnglish
Pages (from-to)340 - 376
JournalGeometric and Functional Analysis
Volume15 (2)
Publication statusPublished - Apr 2005

Bibliographical note

Publisher: Birkhauser Verlag Ag
Other identifier: IDS Number: 945QF

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