A note on the Freiman and Balog-Szemeredi-Gowers theorems in finite fields
We obtain quantitative versions of the Balog-Szemeredi-Gowers and Freiman theorems in the model case of a finite field geometry F_2^n, improving the previously known bounds in such theorems. For instance, if A is a subset of F_2^n such that |A+A| <= K|A| (thus A has small additive doubling),...
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| Format: | Journal article |
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2007
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| _version_ | 1797057793449000960 |
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| author | Green, B Tao, T |
| author_facet | Green, B Tao, T |
| author_sort | Green, B |
| collection | OXFORD |
| description | We obtain quantitative versions of the Balog-Szemeredi-Gowers and Freiman theorems in the model case of a finite field geometry F_2^n, improving the previously known bounds in such theorems. For instance, if A is a subset of F_2^n such that |A+A| <= K|A| (thus A has small additive doubling), we show that there exists an affine subspace V of F_2^n of cardinality |V| >> K^{-O(\sqrt{K})} |A| such that |A \cap V| >> |V|/2K. Under the assumption that A contains at least |A|^3/K quadruples with a_1 + a_2 + a_3 + a_4 = 0 we obtain a similar result, albeit with the slightly weaker condition |V| >> K^{-O(K)}|A|. |
| first_indexed | 2024-03-06T19:41:27Z |
| format | Journal article |
| id | oxford-uuid:20d07796-9535-4ff6-b37b-0299f57c3e2a |
| institution | University of Oxford |
| last_indexed | 2024-03-06T19:41:27Z |
| publishDate | 2007 |
| record_format | dspace |
| spelling | oxford-uuid:20d07796-9535-4ff6-b37b-0299f57c3e2a2022-03-26T11:29:40ZA note on the Freiman and Balog-Szemeredi-Gowers theorems in finite fieldsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:20d07796-9535-4ff6-b37b-0299f57c3e2aSymplectic Elements at Oxford2007Green, BTao, TWe obtain quantitative versions of the Balog-Szemeredi-Gowers and Freiman theorems in the model case of a finite field geometry F_2^n, improving the previously known bounds in such theorems. For instance, if A is a subset of F_2^n such that |A+A| <= K|A| (thus A has small additive doubling), we show that there exists an affine subspace V of F_2^n of cardinality |V| >> K^{-O(\sqrt{K})} |A| such that |A \cap V| >> |V|/2K. Under the assumption that A contains at least |A|^3/K quadruples with a_1 + a_2 + a_3 + a_4 = 0 we obtain a similar result, albeit with the slightly weaker condition |V| >> K^{-O(K)}|A|. |
| spellingShingle | Green, B Tao, T A note on the Freiman and Balog-Szemeredi-Gowers theorems in finite fields |
| title | A note on the Freiman and Balog-Szemeredi-Gowers theorems in finite
fields |
| title_full | A note on the Freiman and Balog-Szemeredi-Gowers theorems in finite
fields |
| title_fullStr | A note on the Freiman and Balog-Szemeredi-Gowers theorems in finite
fields |
| title_full_unstemmed | A note on the Freiman and Balog-Szemeredi-Gowers theorems in finite
fields |
| title_short | A note on the Freiman and Balog-Szemeredi-Gowers theorems in finite
fields |
| title_sort | note on the freiman and balog szemeredi gowers theorems in finite fields |
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