Improved Bounds for Szemerédi's Theorem
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arXiv
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| Format: | Preprint |
| Published: |
2024
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| _version_ | 1866911786150658048 |
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| author | Leng, James Sah, Ashwin Sawhney, Mehtaab |
| author_facet | Leng, James Sah, Ashwin Sawhney, Mehtaab |
| contents | Let $r_k(N)$ denote the size of the largest subset of $[N] = \{1,\ldots,N\}$ with no $k$-term arithmetic progression. We show that for $k\ge 5$, there exists $c_k>0$ such that \[r_k(N)\ll N\exp(-(\log\log N)^{c_k}).\] Our proof is a consequence of recent quasipolynomial bounds on the inverse theorem for the Gowers $U^k$-norm as well as the density increment strategy of Heath-Brown and Szemerédi as reformulated by Green and Tao. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_17995 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Improved Bounds for Szemerédi's Theorem Leng, James Sah, Ashwin Sawhney, Mehtaab Combinatorics Number Theory Let $r_k(N)$ denote the size of the largest subset of $[N] = \{1,\ldots,N\}$ with no $k$-term arithmetic progression. We show that for $k\ge 5$, there exists $c_k>0$ such that \[r_k(N)\ll N\exp(-(\log\log N)^{c_k}).\] Our proof is a consequence of recent quasipolynomial bounds on the inverse theorem for the Gowers $U^k$-norm as well as the density increment strategy of Heath-Brown and Szemerédi as reformulated by Green and Tao. |
| title | Improved Bounds for Szemerédi's Theorem |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2402.17995 |