An inverse theorem for the Gowers U^4 norm
We prove the so-called inverse conjecture for the Gowers U^{s+1}-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we establish that if f : [N] -> C is a function with |f(n)| <= 1 for all n and || f ||_{U^4} >= \delta then there i...
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| Format: | Journal article |
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2009
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| _version_ | 1797089922444689408 |
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| author | Green, B Tao, T Ziegler, T |
| author_facet | Green, B Tao, T Ziegler, T |
| author_sort | Green, B |
| collection | OXFORD |
| description | We prove the so-called inverse conjecture for the Gowers U^{s+1}-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we establish that if f : [N] -> C is a function with |f(n)| <= 1 for all n and || f ||_{U^4} >= \delta then there is a bounded complexity 3-step nilsequence F(g(n)\Gamma) which correlates with f. The approach seems to generalise so as to prove the inverse conjecture for s >= 4 as well, and a longer paper will follow concerning this. By combining this with several previous papers of the first two authors one obtains the generalised Hardy-Littlewood prime-tuples conjecture for any linear system of complexity at most 3. In particular, we have an asymptotic for the number of 5-term arithmetic progressions p_1 < p_2 < p_3 < p_4 < p_5 <= N of primes. |
| first_indexed | 2024-03-07T03:11:00Z |
| format | Journal article |
| id | oxford-uuid:b433c034-6ad3-40d1-a03d-6dfc04a815c2 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T03:11:00Z |
| publishDate | 2009 |
| record_format | dspace |
| spelling | oxford-uuid:b433c034-6ad3-40d1-a03d-6dfc04a815c22022-03-27T04:24:22ZAn inverse theorem for the Gowers U^4 normJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:b433c034-6ad3-40d1-a03d-6dfc04a815c2Symplectic Elements at Oxford2009Green, BTao, TZiegler, TWe prove the so-called inverse conjecture for the Gowers U^{s+1}-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we establish that if f : [N] -> C is a function with |f(n)| <= 1 for all n and || f ||_{U^4} >= \delta then there is a bounded complexity 3-step nilsequence F(g(n)\Gamma) which correlates with f. The approach seems to generalise so as to prove the inverse conjecture for s >= 4 as well, and a longer paper will follow concerning this. By combining this with several previous papers of the first two authors one obtains the generalised Hardy-Littlewood prime-tuples conjecture for any linear system of complexity at most 3. In particular, we have an asymptotic for the number of 5-term arithmetic progressions p_1 < p_2 < p_3 < p_4 < p_5 <= N of primes. |
| spellingShingle | Green, B Tao, T Ziegler, T An inverse theorem for the Gowers U^4 norm |
| title | An inverse theorem for the Gowers U^4 norm |
| title_full | An inverse theorem for the Gowers U^4 norm |
| title_fullStr | An inverse theorem for the Gowers U^4 norm |
| title_full_unstemmed | An inverse theorem for the Gowers U^4 norm |
| title_short | An inverse theorem for the Gowers U^4 norm |
| title_sort | inverse theorem for the gowers u 4 norm |
| work_keys_str_mv | AT greenb aninversetheoremforthegowersu4norm AT taot aninversetheoremforthegowersu4norm AT zieglert aninversetheoremforthegowersu4norm AT greenb inversetheoremforthegowersu4norm AT taot inversetheoremforthegowersu4norm AT zieglert inversetheoremforthegowersu4norm |