An inverse of Furstenberg's correspondence principle and applications to nice recurrence

TL;DR

This paper establishes an inverse to Furstenberg's correspondence principle, linking measure-preserving systems with subsets of natural numbers, and characterizes nice recurrence sets as nicely intersective, advancing understanding in ergodic theory.

Contribution

It introduces an inverse correspondence principle connecting measure-preserving systems to subsets of natural numbers and characterizes nice recurrence sets as nicely intersective.

Findings

Established an inverse of Furstenberg's correspondence principle.

Characterized nice recurrence sets as nicely intersective.

Partially answered two open questions of Moreira.

Abstract

We prove an inverse of Furstenberg's correspondence principle stating that for all measure preserving systems $(X,\mu,T)$ and $A\subset X$ measurable there exists a set $E \subset \mathbb{N}$ such that \[ \mu\left( \bigcap_{i=1}^k T^{-n_i}A\right) = \lim_{N\to \infty} \frac{\left|\left( \bigcap_{i=1}^k (E-n_i) \right)\cap \{0,\dots,N-1\}\right|}{N}\] 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.