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 contribution | A Szemeredi-type regularity lemma in abelian groups, with applications |
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Original language | English |
Pages (from-to) | 340 - 376 |
Journal | Geometric and Functional Analysis |
Volume | 15 (2) |
Publication status | Published - Apr 2005 |
Bibliographical note
Publisher: Birkhauser Verlag AgOther identifier: IDS Number: 945QF