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

2010/06/01 by Ben Green, Terence Tao, Green, Ben +3 · 8 citations
Mathematics · #Analytic Number Theory Research #Dynamical Systems (math.DS) #FOS: Mathematics #Finite Group Theory Research #Limits and Structures in Graph Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1006.0205

openalex publication_date 2010/06/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this note we announce the proof of the inverse conjecture for the Gowers U^s+1[N]-norm for all s => 3; this is new for s => 4, the cases s = 1,2,3 having been previously established. More precisely we outline a proof (details of which will appear in a forthcoming paper) 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. In particular, one obtains an asymptotic formula for the number of k-term arithmetic progressions p₁ < p₂ < ... < p_k <= N of primes, for every k => 3.

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