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arXiv · 1006.0205

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

Ben Green·Terence Tao·Tamar Ziegler
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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]} => \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. In particular, one obtains an asymptotic formula for the number of k-term arithmetic progressions p_1 < p_2 < ... < p_k <= N of primes, for every k => 3.

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Ben Green, Terence Tao, Tamar Ziegler. 2010-06-01. An inverse theorem for the Gowers U^{s+1}[N]-norm (announcement). https://arxiv.org/abs/1006.0205

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