Consider $\mathscr{F}$ a non-empty set of subsets of $\N$.  An operator $T$ on $X$ satisfies property $\p_{\mathscr{F}}$ if for any $U$ non-empty open set in $X$, there exists $x\in X$ such that $\{n\geq 0: T^nx\in U\}\in \mathscr{F}$. Let $\overline{\mathcal{BD}}$ the collection of sets in $\N$ with positive upper Banach density. Our main result is a characterization of sequence of operators satisfying property $\p_{\overline{\mathcal{BD}}}$, for which we have used a result of Bergelson and McCutcheon in the vein of Szemer\'{e}di's theorem. It turns out that operators having  property $\p_{\overline{\mathcal{BD}}}$ satisfy a kind of recurrence described in terms of essential idempotents of $\beta \N$. We will also discuss the case of weighted backward shifts. Finally, we obtain a characterization of reiteratively hypercyclic operators.