Sets of nice recurrence are partition regular

Abstract

A set of positive integers $R$ is called a set of nice recurrence if for any measure preserving system $(X,μ,T)$, for all measurable $A\subseteq X$, and each $\varepsilon>0$, there exists $n\in R$ such that $μ(A\cap T^{-n}A)\geqslant μ(A)^2 - \varepsilon$. Answering a long-standing question of Bergelson, we show that sets of nice recurrence have the following Ramsey property: any finite colouring of a set of nice recurrence admits a monochromatic set of nice recurrence.