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,μ,T) and A⊂ X measurable, there exists a set E ⊂ N such thatμ( i = 1ᵏ T-nᵢA) = limN→ ∞ |( i = 1ᵏ (E-nᵢ) )∩ \0,,N-1\|Nfor all positive integers k and all n₁,,nₖ ∈ N. As a corollary, we show that a set R⊂ N is a set of nice recurrence if and only if it is nicely intersective. Together, the inverse of Furstenberg's correspondence principle and its corollary partially answer two questions of Moreira.