The Furstenberg-Sárközy theorem for polynomials in one or more prime variables
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917655631364096 |
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| author | Doyle, John R. Rice, Alex |
| author_facet | Doyle, John R. Rice, Alex |
| contents | We establish upper bounds on the size of the largest subset of $\{1,2,\dots,N\}$ lacking nonzero differences of the form $h(p_1,\dots,p_{\ell})$, where $h\in \mathbb{Z}[x_1,\dots,x_{\ell}]$ is a fixed polynomial satisfying appropriate conditions and $p_1,\dots,p_{\ell}$ are prime. The bounds are of the same type as the best-known analogs for unrestricted integer inputs, due to Bloom-Maynard and Arala for $\ell=1$, and to the authors for $\ell \geq 2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_00868 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Furstenberg-Sárközy theorem for polynomials in one or more prime variables Doyle, John R. Rice, Alex Number Theory Combinatorics We establish upper bounds on the size of the largest subset of $\{1,2,\dots,N\}$ lacking nonzero differences of the form $h(p_1,\dots,p_{\ell})$, where $h\in \mathbb{Z}[x_1,\dots,x_{\ell}]$ is a fixed polynomial satisfying appropriate conditions and $p_1,\dots,p_{\ell}$ are prime. The bounds are of the same type as the best-known analogs for unrestricted integer inputs, due to Bloom-Maynard and Arala for $\ell=1$, and to the authors for $\ell \geq 2$. |
| title | The Furstenberg-Sárközy theorem for polynomials in one or more prime variables |
| topic | Number Theory Combinatorics |
| url | https://arxiv.org/abs/2405.00868 |