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arXiv · 1009.3998

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

Ben Green·Terence Tao·Tamar Ziegler
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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]} > δthen there is a bounded-complexity s-step nilsequence F(g(n)Γ) which correlates with f, where the bounds on the complexity and correlation depend only on s and δ. 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.

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Ben Green, Terence Tao, Tamar Ziegler. 2026-04-23. An inverse theorem for the Gowers U^{s+1}[N]-norm. https://arxiv.org/abs/1009.3998

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