On polynomial progressions via transference
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916795052457984 |
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| author | Altman, Daniel Sawhney, Mehtaab |
| author_facet | Altman, Daniel Sawhney, Mehtaab |
| contents | We prove new cases of reasonable bounds for the polynomial Szemerédi theorem both over $\mathbb{Z}/N\mathbb{Z}$ with $N$ prime and over the integers. In particular, we prove reasonable bounds for Szemerédi's theorem in the integers with fixed polynomial common difference. That is, we prove for any polynomial $P(y)\in \mathbb{Z}[y]$ with $P(0) = 0$, that the largest subset $A\subseteq [N]$ avoiding the pattern \[x, x+P(y),\ldots, x+ kP(y)\] has size bounded by $\ll_{P,k}N(\log\log\log N)^{-Ω_{P,k}(1)}.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_13010 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On polynomial progressions via transference Altman, Daniel Sawhney, Mehtaab Number Theory Combinatorics We prove new cases of reasonable bounds for the polynomial Szemerédi theorem both over $\mathbb{Z}/N\mathbb{Z}$ with $N$ prime and over the integers. In particular, we prove reasonable bounds for Szemerédi's theorem in the integers with fixed polynomial common difference. That is, we prove for any polynomial $P(y)\in \mathbb{Z}[y]$ with $P(0) = 0$, that the largest subset $A\subseteq [N]$ avoiding the pattern \[x, x+P(y),\ldots, x+ kP(y)\] has size bounded by $\ll_{P,k}N(\log\log\log N)^{-Ω_{P,k}(1)}.$ |
| title | On polynomial progressions via transference |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2506.13010 |