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

2010/09/21 by Ben Green, Terence Tao, Green, Ben +3 · 1 citation
Engineering · Mathematics · #11B30 #Analytic Number Theory Research #Combinatorics (math.CO) #Dynamical Systems (math.DS) #FOS: Mathematics #Limits and Structures in Graph Theory #graph theory and CDMA systems

paper · pdf · doi:10.48550/arxiv.1009.3998

openalex publication_date 2010/09/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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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• Expander Graphs in Pure and Applied Mathematics

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