Erratum for “An inverse theorem for the Gowers U^{s+1}[N]-norm”

What's new 2024-04-25

The purpose of this post is to report an erratum to the 2012 paper “An inverse theorem for the Gowers ${U^{s+1}[N]}$ -norm” of Ben Green, myself, and Tamar Ziegler (previously discussed in this blog post). The main results of this paper have been superseded with stronger quantitative results, first in work of Manners (using somewhat different methods), and more recently in a remarkable paper of Leng, Sah, and Sawhney which combined the methods of our paper with several new innovations to obtain quite strong bounds (of quasipolynomial type); see also an alternate proof of our main results (again by quite different methods) by Candela and Szegedy. In the course of their work, they discovered some fixable but nontrivial errors in our paper. These (rather technical) issues were already implicitly corrected in this followup work which supersedes our own paper, but for the sake of completeness we are also providing a formal erratum for our original paper, which can be found here. We thank Leng, Sah, and Sawhney for bringing these issues to our attention.

Excluding some minor (mostly typographical) issues which we also have reported in this erratum, the main issues stemmed from a conflation of two notions of a degree ${s}$ filtration

$\displaystyle G = G_0 \geq G_1 \geq \dots \geq G_s \geq G_{s+1} = \{1\}$

of a group ${G}$ , which is a nested sequence of subgroups that obey the relation ${[G_i,G_j] \leq G_{i+j}}$ for all ${i,j}$ . The weaker notion (sometimes known as a prefiltration) permits the group ${G_1}$ to be strictly smaller than ${G_0}$ , while the stronger notion requires ${G_0}$ and ${G_1}$ to equal. In practice, one can often move between the two concepts, as ${G_1}$ is always normal in ${G_0}$ , and a prefiltration behaves like a filtration on every coset of ${G_1}$ (after applying a translation and perhaps also a conjugation). However, we did not clarify this issue sufficiently in the paper, and there are some places in the text where results that were only proven for filtrations were applied for prefiltrations. The erratum fixes this issues, mostly by clarifying that we work with filtrations throughout (which requires some decomposition into cosets in places where prefiltrations are generated). Similar adjustments need to be made for multidegree filtrations and degree-rank filtrations, which we also use heavily on our paper.

In most cases, fixing this issue only required minor changes to the text, but there is one place (Section 8) where there was a non-trivial problem: we used the claim that the final group ${G_s}$ was a central group, which is true for filtrations, but not necessarily for prefiltrations. This fact (or more precisely, a multidegree variant of it) was used to claim a factorization for a certain product of nilcharacters, which is in fact not true as stated. In the erratum, a substitute factorization for a slightly different product of nilcharacters is provided, which is still sufficient to conclude the main result of this part of the paper (namely, a statistical linearization of a certain family of nilcharacters in the shift parameter ${h}$ ).

Again, we stress that these issues do not impact the paper of Leng, Sah, and Sawhney, as they adapted the methods in our paper in a fashion that avoids these errors.