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An inverse theorem for the Gowers \(U^{s+1}[N]\)-norm. (English) Zbl 1282.11007
Authors’ abstract: We prove the inverse conjecture for the Gowers \(U^{s+1}[N]\)-norm for all \(s\geq 1\); this is new for \(s \geq 4\). More precisely, we establish that if \(f: [N] \rightarrow [-1,1]\) is a function with \(\| f \|_{U^{s+1}[N]} \geq \delta\), then there is a bounded-complexity \(s\)-step nilsequence \(F(g(n)\Gamma)\) that 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.

MSC:
11B30 Arithmetic combinatorics; higher degree uniformity
11N05 Distribution of primes
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