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

Abstract.Szemerédi’s regularity lemma is an important tool in graph theory which has applications throughout combinatorics. In this paper we prove an analogue of Szemerédi’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 $$A \subseteq \{1,\ldots,N\}$$ has δ N2 triples (a1, a2, a3) for which a1 + a2 = a3 then A = B ∪ C, where B is sum-free and |C| = δ′N, and $$\delta^{\prime} \rightarrow 0$$ as $$\delta \rightarrow 0.$$ Another answers a question of Bergelson, Host and Kra. If $$\alpha, \epsilon > 0,$$ if $$N\,>\,N_{0}(\alpha, \epsilon)$$ and if $$A \subseteq \{1,\ldots,N\}$$ has size α N, then there is some d ≠ 0 such that A contains at least $$(\alpha^{3}-\epsilon)N$$ three-term arithmetic progressions with common difference d.

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