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