## 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

### Keywords:

equidistribution; Gowers norms; nilmanifold; nilsequences

Zbl 1251.37012
Full Text:

### 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
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