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

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 $$\mu \left(\bigcap _{i=1}^k{T}^{-n_i}A\right)=\underset{N\to \mathrm{\infty }}{\lim }\frac{\mid (\bigcap _{i=1}^k(E-n_i))\cap \{0,\dots ,N-1\}\mid }{N}$$ 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.