##
**On the quantitative distribution of polynomial nilsequences – erratum.**
*(English)*
Zbl 1290.37004

Summary: This is an erratum to the authors’ paper [ibid. 175, No. 2, 465–540 (2012; Zbl 1251.37012)]. 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 \(N_1 = \cdots = N_t = N\) 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.

### MSC:

37A45 | Relations of ergodic theory with number theory and harmonic analysis (MSC2010) |

11N13 | Primes in congruence classes |

### Citations:

Zbl 1251.37012
PDF
BibTeX
XML
Cite

\textit{B. Green} and \textit{T. Tao}, Ann. Math. (2) 179, No. 3, 1175--1183 (2014; Zbl 1290.37004)

### References:

[1] | B. Green and T. Tao, ”The quantitative behaviour of polynomial orbits on nilmanifolds,” Ann. of Math., vol. 175, iss. 2, pp. 465-540, 2012. · Zbl 1251.37012 |

[2] | 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. · Zbl 1338.37040 |

[3] | A. Gorodnik and R. Spatzier, Exponential Mixing of Nilmanifold Automorphisms. · Zbl 1312.37007 |

[4] | A. Gorodnik and R. Spatzier, Mixing properties of commuting nilmanifold automorphisms. · Zbl 1312.37007 |

[5] | 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. · Zbl 1222.11015 |

[6] | 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. · Zbl 1282.11007 |

[7] | B. Green and T. Tao, On the quantitative distribution of polynomial nilsequences - erratum. · Zbl 1290.37004 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.