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An Irregular Mind pp 261–334Cite as

An Arithmetic Regularity Lemma, An Associated Counting Lemma, and Applications

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Part of the book series: Bolyai Society Mathematical Studies ((BSMS,volume 21))

Abstract

To Endre Szemerédi on the occasion of his 70th birthday Szemerédi’s regularity lemma can be viewed as a rough structure theorem for arbitrary dense graphs, decomposing such graphs into a structured piece, a small error, and a uniform piece. We establish an arithmetic regularity lemma that similarly decomposes bounded functions f: [N] →ℂ, into a (well-equidistributed, virtual) s-step nilsequence, an error which is small in L2 and a further error which is minuscule in the Gowers Us+1-norm, where s ≥ 1 is a parameter. We then establish a complementary arithmetic counting lemma that counts arithmetic patterns in the nilsequence component of f.

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Authors and Affiliations

  1. Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CBS 0WA, England

    Ben Green

  2. Department of Mathematics UCLA, Los Angeles, CA, 90095, USA

    Terence Tao

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  1. Ben Green
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  2. Terence Tao
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Correspondence to Ben Green .

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Editors and Affiliations

  1. Alfred Rènyi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, Budapest, 1053, Hungary

    Imre Bárány

  2. Department of Mathematics, University of British Columbia, Mathematics Road 1984, V6T 1Z2, Vancouver, British Columbia, Canada

    József Solymosi

  3. Alfred Rènyi Institute of Mathematics, Hungarian Academy of Sciences, Reáltanoda u. 13-15, Budapest, 1053, Hungary

    Gábor Sági

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© 2010 János Bolyai Mathematical Society and Springer-Verlag

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Green, B., Tao, T. (2010). An Arithmetic Regularity Lemma, An Associated Counting Lemma, and Applications. In: Bárány, I., Solymosi, J., Sági, G. (eds) An Irregular Mind. Bolyai Society Mathematical Studies, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14444-8_7

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