Hostname: page-component-5447f9dfdb-lxcff Total loading time: 0 Render date: 2025-07-29T14:28:10.851Z Has data issue: false hasContentIssue false
  • English
  • Français

A non-flag arithmetic regularity lemma and counting lemma

Part of: Sequences and sets

Published online by Cambridge University Press:  02 July 2025

Daniel Altman
Daniel Altman*
Affiliation:
Department of Mathematics, Sloan Mathematical Center, Stanford University, 450 Jane Stanford Way Building 380, Stanford, CA 94305, USA daniel.h.altman@gmail.com
Get access Rights & Permissions [Opens in a new window]

Abstract

Green and Tao’s arithmetic regularity lemma and counting lemma together apply to systems of linear forms which satisfy a particular algebraic criterion known as the ‘flag condition’. We give an arithmetic regularity lemma and counting lemma which apply to all systems of linear forms.

Keywords

regularity lemmanilsequencecounting lemma

MSC classification

Primary: 11B30: Arithmetic combinatorics; higher degree uniformity

Information

Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 3 , March 2025 , pp. 635 - 679
Check for updates
Copyright
© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence.

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

Altman, D., On a conjecture of Gowers and Wolf, Discrete Anal. (2022), 10.Google Scholar
Cohen, H., A course in computational algebraic number theory , Graduate Texts in Mathematics, vol. 138 (Springer, Berlin, 1993).Google Scholar
Corwin, L. J. and Greenleaf, F. P., Representations of nilpotent Lie groups and their applications. Part I: Basic theory and examples, Cambridge Studies in Advanced Mathematics, vol. 18 (Cambridge University Press, Cambridge, 1990).Google Scholar
Gowers, W. T. and Wolf, J., The true complexity of a system of linear equations, Proc. Lond. Math. Soc. (3) 100 (2010), 155–176.Google Scholar
Green, B., A Szemerédi-type regularity lemma in abelian groups, with applications, Geom. Funct. Anal. 15 (2005), 340–376.10.1007/s00039-005-0509-8CrossRefGoogle Scholar
Green, B. and Tao, T., An arithmetic regularity lemma, an associated counting lemma, and applications, in An irregular mind, Bolyai Society Mathematical Studies, vol. 21 (János Bolyai Mathematical Society, Budapest, 2010), 261–334.Google Scholar
Green, B. and Tao, T., The quantitative behaviour of polynomial orbits on nilmanifolds, Ann. of Math. (2) 175 (2012), 465–540.Google Scholar
Green, B. and Tao, T., On the quantitative distribution of polynomial nilsequences–erratum, Preprint (2015), arxiv:1311.6170.Google Scholar
Green, B. and Tao, T., An arithmetic regularity lemma, associated counting lemma, and applications, Preprint (2020), arXiv:1002.2028v3.Google Scholar
Hall, B. C., Lie groups, Lie algebras, and representations: An elementary introduction, Graduate Texts in Mathematics (Springer, Cham, 2003).10.1007/978-0-387-21554-9CrossRefGoogle Scholar
Kirillov, A. Jr., An introduction to Lie groups and Lie algebras, Cambridge Studies in Advanced Mathematics (Cambridge University Press, Cambridge, 2008).Google Scholar
Leibman, A., Polynomial mappings of groups, Israel J. Math. 129 (2002), 29–60.10.1007/BF02773152CrossRefGoogle Scholar
Leibman, A., Pointwise convergence of ergodic averages for polynomial actions of ${\Bbb Z}^d$ by translations on a nilmanifold, Ergodic Theory Dynam. Systems 25 (2005), 215–225.10.1017/S0143385704000227CrossRefGoogle Scholar
Manners, F., True complexity and iterated Cauchy–Schwarz, Preprint (2021), arXiv:2109.05731.Google Scholar
Szemerédi, E., On sets of integers containing no k elements in arithmetic progression, Acta Arith. 27 (1975), 199–245.10.4064/aa-27-1-199-245CrossRefGoogle Scholar
Tao, T., A correction to “An arithmetic regularity lemma, an associated counting lemma, and applications” (2020), https://terrytao.wordpress.com/2020/11/26/a-correction-toan-arithmetic-regularity-lemma-an-associated-counting-lemma-and-applications/.Google Scholar