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arXiv · 2410.14427

Additive Ramsey theory over Piatetski-Shapiro numbers

Jonathan Chapman·Sam Chow·Philippa Holdridge
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Abstract

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.

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Jonathan Chapman, Sam Chow, Philippa Holdridge. 2025-03-20. Additive Ramsey theory over Piatetski-Shapiro numbers. https://arxiv.org/abs/2410.14427

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