Additive Ramsey theory over Piatetski-Shapiro numbers

TL;DR

This paper characterizes when linear equations are partition regular over Piatetski-Shapiro numbers for certain exponents and establishes related density results, updating a key Fourier-analytic transference principle.

Contribution

It provides new characterizations of partition regularity over Piatetski-Shapiro numbers and updates a Fourier-analytic transference principle with strengthened conclusions.

Findings

Partition regularity characterized for specific exponents c

Density results with quantitative bounds established

Updated Fourier-analytic transference principle with stronger conclusions

Abstract

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