On the quantitative distribution of polynomial nilsequences - erratum

TL;DR

This erratum addresses a proof error in a previous paper on polynomial nilsequences, providing fixes for the equal sides case and sketching a proof for the general multiparameter case.

Contribution

The paper corrects and extends previous results on the distribution of polynomial nilsequences, including a new proof for the multiparameter case.

Findings

The equal sides case of the distribution theorem is derived from 1-parameter results.

A sketch of proof for the full multiparameter distribution theorem is provided.

The original proof's error is addressed with new arguments.

Abstract

This is an erratum to 'On the quantitative distribution of polynomial nilsequences' [GT]. 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 $N_1 = \dots = N_t = N$ of [GT, Theorem 8.6] from the 1-parameter results in [GT]. This is the same basic mode of argument we attempted in the original paper, 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. Second, we sketch a proof that [GT, Theorem 8.6] does in fact hold in its originally stated form, that is to say without the equal sides condition. To obtain this statement the entire argument of [GT] must be run 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.