An inverse of Furstenberg's correspondence principle and applications to nice recurrence
Fish, Alexander, Sean Skinner · arXiv (Cornell University) · 2024
We prove an inverse of Furstenberg's correspondence principle stating that for all measure preserving systems $(X,μ,T)$ and $A\subset X$ measurable there exists a set $E \subset \mathbb{N}$ such that \[ μ\left( \bigcap_{i=1}^k T^{-n_i}A\right) = \lim_{N\to \infty} \frac{\left|\left( \bigcap_{i=1}^k (E-n_i) \right)\cap \{0,\dots,N-1\}\right|}{N}\] for all $k,n_1,\dots,n_k \in \mathbb{N}$. As a corollary we show that a set $R\subset \mathbb{N}$ is a set of nice recurrence if and only if it is nicely intersective. Together, the inverse of Furstenberg's correspondence principle and it's corollary partially answer two questions of Moreira.