An inverse theorem for the Gowers U^{s+1}[N]-norm
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2010
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| _version_ | 1866915951128084480 |
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| author | Green, Ben Tao, Terence Ziegler, Tamar |
| author_facet | Green, Ben Tao, Terence Ziegler, Tamar |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1009_3998 |
| institution | arXiv |
| publishDate | 2010 |
| record_format | arxiv |
| spellingShingle | An inverse theorem for the Gowers U^{s+1}[N]-norm Green, Ben Tao, Terence Ziegler, Tamar Combinatorics Dynamical Systems 11B30 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. |
| title | An inverse theorem for the Gowers U^{s+1}[N]-norm |
| topic | Combinatorics Dynamical Systems 11B30 |
| url | https://arxiv.org/abs/1009.3998 |