An equivalence between inverse sumset theorems and inverse conjectures for the U^3 norm
We establish a correspondence between inverse sumset theorems (which can be viewed as classifications of approximate (abelian) groups) and inverse theorems for the Gowers norms (which can be viewed as classifications of approximate polynomials). In particular, we show that the inverse sumset theorem...
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| Format: | Journal article |
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2009
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| _version_ | 1797050938811219968 |
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| author | Green, B Tao, T |
| author_facet | Green, B Tao, T |
| author_sort | Green, B |
| collection | OXFORD |
| description | We establish a correspondence between inverse sumset theorems (which can be viewed as classifications of approximate (abelian) groups) and inverse theorems for the Gowers norms (which can be viewed as classifications of approximate polynomials). In particular, we show that the inverse sumset theorems of Freiman type are equivalent to the known inverse results for the Gowers U^3 norms, and moreover that the conjectured polynomial strengthening of the former is also equivalent to the polynomial strengthening of the latter. We establish this equivalence in two model settings, namely that of the finite field vector spaces F_2^n, and of the cyclic groups Z/NZ. In both cases the argument involves clarifying the structure of certain types of approximate homomorphism. |
| first_indexed | 2024-03-06T18:12:38Z |
| format | Journal article |
| id | oxford-uuid:03892663-c480-46fc-b7e4-368a6bb39e9d |
| institution | University of Oxford |
| last_indexed | 2024-03-06T18:12:38Z |
| publishDate | 2009 |
| record_format | dspace |
| spelling | oxford-uuid:03892663-c480-46fc-b7e4-368a6bb39e9d2022-03-26T08:46:50ZAn equivalence between inverse sumset theorems and inverse conjectures for the U^3 normJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:03892663-c480-46fc-b7e4-368a6bb39e9dSymplectic Elements at Oxford2009Green, BTao, TWe establish a correspondence between inverse sumset theorems (which can be viewed as classifications of approximate (abelian) groups) and inverse theorems for the Gowers norms (which can be viewed as classifications of approximate polynomials). In particular, we show that the inverse sumset theorems of Freiman type are equivalent to the known inverse results for the Gowers U^3 norms, and moreover that the conjectured polynomial strengthening of the former is also equivalent to the polynomial strengthening of the latter. We establish this equivalence in two model settings, namely that of the finite field vector spaces F_2^n, and of the cyclic groups Z/NZ. In both cases the argument involves clarifying the structure of certain types of approximate homomorphism. |
| spellingShingle | Green, B Tao, T An equivalence between inverse sumset theorems and inverse conjectures for the U^3 norm |
| title | An equivalence between inverse sumset theorems and inverse conjectures
for the U^3 norm |
| title_full | An equivalence between inverse sumset theorems and inverse conjectures
for the U^3 norm |
| title_fullStr | An equivalence between inverse sumset theorems and inverse conjectures
for the U^3 norm |
| title_full_unstemmed | An equivalence between inverse sumset theorems and inverse conjectures
for the U^3 norm |
| title_short | An equivalence between inverse sumset theorems and inverse conjectures
for the U^3 norm |
| title_sort | equivalence between inverse sumset theorems and inverse conjectures for the u 3 norm |
| work_keys_str_mv | AT greenb anequivalencebetweeninversesumsettheoremsandinverseconjecturesfortheu3norm AT taot anequivalencebetweeninversesumsettheoremsandinverseconjecturesfortheu3norm AT greenb equivalencebetweeninversesumsettheoremsandinverseconjecturesfortheu3norm AT taot equivalencebetweeninversesumsettheoremsandinverseconjecturesfortheu3norm |