On the quantitative distribution of polynomial nilsequences — erratum

Abstract

This is an erratum to the paper The quantitative behaviour of polynomial orbits on nilmanifolds by the authors, published as Ann. of Math. (2) $\bf{175}$ (2012), no. 2, 465–540. The proof of Theorem 8.6 of that paper, which claims a distribution result for multiparameter polynomial sequences on nilmanifolds, was incorrect. We provide two fixes for this issue here. First, we deduce the “equal sides” case $N_1 = …= N_t = N$ of this result from the 1-parameter results in the paper. This is the same basic mode of argument we attempted originally, though the details are different. The equal sides case is the only one required in applications such as the proof of the inverse conjectures for the Gowers norms due to the authors and Ziegler. To remove the equal sides condition one must rerun the entire argument of our paper in the context of multiparameter polynomial sequences $g : \mathbb{Z}^t \rightarrow G$ rather than 1-parameter sequences $g : \mathbb{Z} \rightarrow G$ as is currently done: a more detailed sketch of how this may be done is available online.

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  • [green-tao-nilratner] Go to document B. Green and T. Tao, "The quantitative behaviour of polynomial orbits on nilmanifolds," Ann. of Math., vol. 175, iss. 2, pp. 465-540, 2012.
    @article {green-tao-nilratner, MRKEY = {2877065},
      AUTHOR = {Green, Ben and Tao, Terence},
      TITLE = {The quantitative behaviour of polynomial orbits on nilmanifolds},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {175},
      YEAR = {2012},
      NUMBER = {2},
      PAGES = {465--540},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {37A15},
      MRNUMBER = {2877065},
      MRREVIEWER = {Tamar Ziegler},
      ZBLNUMBER = {06024997},
      DOI = {10.4007/annals.2012.175.2.2},
      }
  • [fks] Go to document D. Fisher, B. Kalinin, and R. Spatzier, "Global rigidity of higher rank Anosov actions on tori and nilmanifolds," J. Amer. Math. Soc., vol. 26, iss. 1, pp. 167-198, 2013.
    @article {fks, MRKEY = {2983009},
      AUTHOR = {Fisher, David and Kalinin, Boris and Spatzier, Ralf},
      TITLE = {Global rigidity of higher rank {A}nosov actions on tori and nilmanifolds},
      NOTE = {with an appendix by James F. Davis},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {26},
      YEAR = {2013},
      NUMBER = {1},
      PAGES = {167--198},
      ISSN = {0894-0347},
      MRCLASS = {37Dxx (42B05)},
      MRNUMBER = {2983009},
      ZBLNUMBER = {06168120},
      DOI = {10.1090/S0894-0347-2012-00751-6},
      }
  • [gorodnik-spatzier] A. Gorodnik and R. Spatzier, Exponential Mixing of Nilmanifold Automorphisms.
    @misc{gorodnik-spatzier,
      author={A.~Gorodnik and R.~Spatzier},
      TITLE={Exponential Mixing of Nilmanifold Automorphisms},
      NOTE={to appear in \emph{J. Anal. Math.}},
      ARXIV = {1210.2271},
      }
  • [gorodnik-spatzier2] A. Gorodnik and R. Spatzier, Mixing properties of commuting nilmanifold automorphisms.
    @misc{gorodnik-spatzier2,
      author={A.~Gorodnik and R.~Spatzier},
      TITLE={Mixing properties of commuting nilmanifold automorphisms},
      ARXIV={1211.0987},
      }
  • [arithmetic-regularity] Go to document B. Green and T. Tao, "An arithmetic regularity lemma, an associated counting lemma, and applications," in An Irregular Mind, Budapest: János Bolyai Math. Soc., 2010, vol. 21, pp. 261-334.
    @incollection {arithmetic-regularity, MRKEY = {2815606},
      AUTHOR = {Green, Ben and Tao, Terence},
      TITLE = {An arithmetic regularity lemma, an associated counting lemma, and applications},
      BOOKTITLE = {An Irregular Mind},
      SERIES = {Bolyai Soc. Math. Stud.},
      VOLUME = {21},
      PAGES = {261--334},
      PUBLISHER = {János Bolyai Math. Soc.},
      ADDRESS = {Budapest},
      YEAR = {2010},
      MRCLASS = {11B30 (05D05)},
      MRNUMBER = {2815606},
      MRREVIEWER = {David Conlon},
      ZBLNUMBER = {1222.11015},
      DOI = {10.1007/978-3-642-14444-8_7},
      }
  • [gtz] Go to document B. Green, T. Tao, and T. Ziegler, "An inverse theorem for the Gowers $U^{s+1}[N]$-norm," Ann. of Math., vol. 176, iss. 2, pp. 1231-1372, 2012.
    @article {gtz, MRKEY = {2950773},
      AUTHOR = {Green, Ben and Tao, Terence and Ziegler, Tamar},
      TITLE = {An inverse theorem for the {G}owers {$U\sp {s+1}[N]$}-norm},
      JOURNAL = {Ann. of Math.},
      FJOURNAL = {Annals of Mathematics. Second Series},
      VOLUME = {176},
      YEAR = {2012},
      NUMBER = {2},
      PAGES = {1231--1372},
      ISSN = {0003-486X},
      CODEN = {ANMAAH},
      MRCLASS = {11B30},
      MRNUMBER = {2950773},
      MRREVIEWER = {Julia Wolf},
      ZBLNUMBER = {06093950},
      DOI = {10.4007/annals.2012.176.2.11},
      }
  • [more-detailed] B. Green and T. Tao, On the quantitative distribution of polynomial nilsequences — erratum.
    @misc{more-detailed,
      author={Green, Ben and Tao, Terence},
      TITLE={On the quantitative distribution of polynomial nilsequences -- erratum},
      NOTE={not for publication},
      SORTYEAR={2015},
      ARXIV={1311.6170},
     }

Authors

Ben Green

Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, England

Terence Tao

Department of Mathematics, University of California Los Angeles, 405 Hilgard Ave., Los Angeles, CA 90095