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| Main Authors: | , |
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| Format: | Preprint |
| Izdano: |
2006
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| Teme: | |
| Online dostop: | https://arxiv.org/abs/math/0610604 |
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| _version_ | 1866911747627024384 |
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| author | Green, Ben Tao, Terence |
| author_facet | Green, Ben Tao, Terence |
| contents | Define $r_4(N)$ to be the largest cardinality of a set $A$ in $\{1,\dots,N\}$ which does not contain four elements in arithmetic progression. In 1998 Gowers proved that $r_4(N) \ll N(\log \log N)^{-c}$ for some absolute constant $c> 0$. In this paper (part II of a series) we improve this to $r_4(N) \ll N e^{-c\sqrt{\log \log N}}$. In part III of the series we will use a more elaborate argument to improve this to $r_4(N) \ll N(\log N)^{-c}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_math_0610604 |
| institution | arXiv |
| publishDate | 2006 |
| record_format | arxiv |
| spellingShingle | New bounds for Szemeredi's theorem, II: A new bound for $r_4(N)$ Green, Ben Tao, Terence Number Theory Combinatorics Define $r_4(N)$ to be the largest cardinality of a set $A$ in $\{1,\dots,N\}$ which does not contain four elements in arithmetic progression. In 1998 Gowers proved that $r_4(N) \ll N(\log \log N)^{-c}$ for some absolute constant $c> 0$. In this paper (part II of a series) we improve this to $r_4(N) \ll N e^{-c\sqrt{\log \log N}}$. In part III of the series we will use a more elaborate argument to improve this to $r_4(N) \ll N(\log N)^{-c}$. |
| title | New bounds for Szemeredi's theorem, II: A new bound for $r_4(N)$ |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/math/0610604 |