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.