## 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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### References:

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