An arithmetic regularity lemma, associated counting lemma, and applications

Details An arithmetic regularity lemma, associated counting lemma, and applications An arithmetic regularity lemma, associated counting lemma, and applications

2010-02-10

arxiv.org/abs/1002.2028v2

Mathematics

Combinatorics

60 pages, to appear in Proceedings in honour of the 70th birthday of Endre Szemeredi

Scientific paper

Szemeredi's regularity lemma can be viewed as a rough structure theorem for arbitrary dense graphs, decomposing such graphs into a structured piece (a partition into cells with edge densities), a small error (corresponding to irregular cells), and a uniform piece (the pseudorandom deviations from the edge densities). We establish an arithmetic regularity lemma that similarly decomposes bounded functions f : [N] -> C, into a (well-equidistributed, virtual) -step nilsequence, an error which is small in L^2 and a further error which is miniscule in the Gowers U^{s+1}-norm, where s is a positive integer. We then establish a complementary arithmetic counting lemma that counts arithmetic patterns in the nilsequence component of f. We provide a number of applications of these lemmas: a proof of Szemeredi's theorem on arithmetic progressions, a proof of a conjecture of Bergelson, Host and Kra, and a generalisation of certain results of Gowers and Wolf. Our result is dependent on the inverse conjecture for the Gowers U^{s+1} norm, recently established for general s by the authors and T. Ziegler.

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