Additive Ramsey theory over Piatetski-Shapiro numbers

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