Additive Ramsey theory over Piatetski-Shapiro numbers
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866917491808141312 |
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| author | Chapman, Jonathan Chow, Sam Holdridge, Philippa |
| author_facet | Chapman, Jonathan Chow, Sam Holdridge, Philippa |
| contents | We characterise partition regularity for linear equations over the Piatetski-Shapiro numbers $\lfloor n^c \rfloor$ when $1 < c < c^†(s)$, where $s \geqslant 3$ is the number of variables. Here $c^†(3) = 12/11$ and $c^†(4) = 7/6$, while $c^†(s) = 2$ for $s \geqslant 5$. We also establish density results with quantitative bounds. Following recent developments, we take this opportunity to update Browning and Prendiville's version of Green's Fourier-analytic transference principle, strengthening its conclusion. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_14427 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Additive Ramsey theory over Piatetski-Shapiro numbers Chapman, Jonathan Chow, Sam Holdridge, Philippa Number Theory Combinatorics 11B30 (Primary), 05D10, 11D72, 11L15 (Secondary) We characterise partition regularity for linear equations over the Piatetski-Shapiro numbers $\lfloor n^c \rfloor$ when $1 < c < c^†(s)$, where $s \geqslant 3$ is the number of variables. Here $c^†(3) = 12/11$ and $c^†(4) = 7/6$, while $c^†(s) = 2$ for $s \geqslant 5$. We also establish density results with quantitative bounds. Following recent developments, we take this opportunity to update Browning and Prendiville's version of Green's Fourier-analytic transference principle, strengthening its conclusion. |
| title | Additive Ramsey theory over Piatetski-Shapiro numbers |
| topic | Number Theory Combinatorics 11B30 (Primary), 05D10, 11D72, 11L15 (Secondary) |
| url | https://arxiv.org/abs/2410.14427 |