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 $s \geqslant 3$ 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&#39;s version of Green&#39;s Fourier-analytic transference principle, strengthening its conclusion.