An inverse theorem for the Gowers U^3 norm
The Gowers U^3 norm is one of a sequence of norms used in the study of arithmetic progressions. If G is an abelian group and A is a subset of G then the U^3(G) of the characteristic function 1_A is useful in the study of progressions of length 4 in A. We give a comprehensive study of the U^3(G) norm...
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| Format: | Journal article |
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2005
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| _version_ | 1826297326823538688 |
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| author | Green, B Tao, T |
| author_facet | Green, B Tao, T |
| author_sort | Green, B |
| collection | OXFORD |
| description | The Gowers U^3 norm is one of a sequence of norms used in the study of arithmetic progressions. If G is an abelian group and A is a subset of G then the U^3(G) of the characteristic function 1_A is useful in the study of progressions of length 4 in A. We give a comprehensive study of the U^3(G) norm, obtaining a reasonably complete description of functions f : G -> C for which ||f||_{U^3} is large and providing links to recent results of Host, Kra and Ziegler in ergodic theory. As an application we generalise a result of Gowers on Szemeredi's theorem. Writing r_4(G) for the size of the largest set A not containing four distinct elements in arithmetic progression, we show that r_4(G) << |G|(loglog|G|)^{-c} for some absolute constant c. In future papers we will develop these ideas further, obtaining an asymptotic for the number of 4-term progressions p_1 < p_2 < p_3 < p_4 < N of primes as well as superior bounds for r_4(G). |
| first_indexed | 2024-03-07T04:29:53Z |
| format | Journal article |
| id | oxford-uuid:cdf11dfd-9ac7-4d3e-90e7-65df5b429306 |
| institution | University of Oxford |
| last_indexed | 2024-03-07T04:29:53Z |
| publishDate | 2005 |
| record_format | dspace |
| spelling | oxford-uuid:cdf11dfd-9ac7-4d3e-90e7-65df5b4293062022-03-27T07:32:16ZAn inverse theorem for the Gowers U^3 normJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:cdf11dfd-9ac7-4d3e-90e7-65df5b429306Symplectic Elements at Oxford2005Green, BTao, TThe Gowers U^3 norm is one of a sequence of norms used in the study of arithmetic progressions. If G is an abelian group and A is a subset of G then the U^3(G) of the characteristic function 1_A is useful in the study of progressions of length 4 in A. We give a comprehensive study of the U^3(G) norm, obtaining a reasonably complete description of functions f : G -> C for which ||f||_{U^3} is large and providing links to recent results of Host, Kra and Ziegler in ergodic theory. As an application we generalise a result of Gowers on Szemeredi's theorem. Writing r_4(G) for the size of the largest set A not containing four distinct elements in arithmetic progression, we show that r_4(G) << |G|(loglog|G|)^{-c} for some absolute constant c. In future papers we will develop these ideas further, obtaining an asymptotic for the number of 4-term progressions p_1 < p_2 < p_3 < p_4 < N of primes as well as superior bounds for r_4(G). |
| spellingShingle | Green, B Tao, T An inverse theorem for the Gowers U^3 norm |
| title | An inverse theorem for the Gowers U^3 norm |
| title_full | An inverse theorem for the Gowers U^3 norm |
| title_fullStr | An inverse theorem for the Gowers U^3 norm |
| title_full_unstemmed | An inverse theorem for the Gowers U^3 norm |
| title_short | An inverse theorem for the Gowers U^3 norm |
| title_sort | inverse theorem for the gowers u 3 norm |
| work_keys_str_mv | AT greenb aninversetheoremforthegowersu3norm AT taot aninversetheoremforthegowersu3norm AT greenb inversetheoremforthegowersu3norm AT taot inversetheoremforthegowersu3norm |