An inverse theorem for the Gowers U^{s+1}[N]-norm
Abstract
We prove the inverse conjecture for the Gowers U^{s+1}[N]-norm for all s >= 3; this is new for s > 3, and the cases s<3 have also been previously established. More precisely, we establish that if f : [N] -> [-1,1] is a function with || f ||_{U^{s+1}[N]} > \delta then there is a bounded-complexity s-step nilsequence F(g(n)\Gamma) which correlates with f, where the bounds on the complexity and correlation depend only on s and \delta. From previous results, this conjecture implies the Hardy-Littlewood prime tuples conjecture for any linear system of finite complexity. A 6-page erratum to the original paper was provided in April 2024 and is available as a separate PDF on the webpages of the first and second authors.
Keywords
Cite
@article{arxiv.1009.3998,
title = {An inverse theorem for the Gowers U^{s+1}[N]-norm},
author = {Ben Green and Terence Tao and Tamar Ziegler},
journal= {arXiv preprint arXiv:1009.3998},
year = {2026}
}
Comments
Appeared as Annals of Math. 176 (2012), 1231--1372. Preprint is 116 pages. This version corrects a mistake in the proof of Lemma 13.2 (the wrong filtration was specified) and two small typos on page 28