Additive Ramsey theory over Piatetski-Shapiro numbers

Abstract

We characterise partition regularity for linear equations over the Piatetski-Shapiro numbers $\lfloor n^c\rfloor$ when $1<c<c^{†}(s)$, where $s⩾3$ is the number of variables. Here $c^{†}(3)=12/11$ and $c^{†}(4)=7/6$, while $c^{†}(s)=2$ for $s⩾5$. 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.

arXiv comment20 pages; added a new section on updating the transference principle, which is then used to improve the quantitative density bounds in Theorem 1.8