Additive Ramsey theory over Piatetski-Shapiro numbers
Jonathan Chapman, Sam Chow, Philip Holdridge
TL;DR
This work analyzes partition regularity and density phenomena for linear equations over the Piatetski-Shapiro numbers $PS_c$ in the regime $1<c<c^{\dagger}(s)$, with explicit threshold values $c^{\dagger}(3)=12/11$, $c^{\dagger}(4)=7/6$, and $c^{\dagger}(s)=2$ for $s\ge 5$. It advances a Fourier-analytic transference framework by refining decay and restriction estimates and by incorporating the latest pseudorandomness results, enabling quantitative density bounds and monochromatic solutions to $\mathbf c\cdot\mathbf x=0$ with distinct coordinates in colourings of $PS_c(N)$. The paper develops a weighted counting method using $\nu$ supported on $PS_c(N)$, establishes a strong Fourier decay $\|\hat{\nu}-\hat{1}_{[N]}\|_{\infty}$ bound, and proves two restriction estimates, which feed into an updated transference principle that avoids the $W$-trick. With these tools, it proves both a Ramsey-type result (existence of monochromatic solutions) and a quantitative density-regularity bound $|A| \le |PS_c(N)| \exp(-C^{-1}(\log\log N)^{1/\tau(s)})$ for appropriate $\tau(s)$, illustrating substantial density savings for PS_c-avoiding sets. These results sharpen our understanding of nonlinear Diophantine phenomena in irregular sequences and broaden the reach of Green’s transference method in additive combinatorics.
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.