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Abstract: Szemeredi-s regularity lemma can be viewed as a rough structure theorem forarbitrary dense graphs, decomposing such graphs into a structured piece apartition into cells with edge densities, a small error corresponding toirregular cells, and a uniform piece the pseudorandom deviations from theedge densities. We establish an arithmetic regularity lemma that similarlydecomposes bounded functions f : N -> C, into a well-equidistributed,virtual -step nilsequence, an error which is small in L^2 and a further errorwhich is miniscule in the Gowers U^{s+1}-norm, where s is a positive integer.We then establish a complementary arithmetic counting lemma that countsarithmetic patterns in the nilsequence component of f.We provide a number of applications of these lemmas: a proof of Szemeredi-stheorem on arithmetic progressions, a proof of a conjecture of Bergelson, Hostand 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.



Author: Ben Green, Terence Tao

Source: https://arxiv.org/







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