  We generalize the notions of hypercyclic operators, $\mathfrak{U}$-frequently
hypercyclic operators and frequently hypercyclic operators by introducing a new
notion of hypercyclicity, called $\mathcal{A}$-frequent hypercyclicity. We then
state an $\mathcal{A}$-Frequent Hypercyclicity Criterion, inspired from the
Hypercyclicity Criterion and the Frequent Hypercyclicity Criterion, and we show
that this criterion characterizes the $\mathcal{A}$-frequent hypercyclicity for
weighted shifts. We finish by investigating which kind of properties of density
can have the sets ${N(x, U)=\{n\in \mathbb{N}:T^nx\in U\}}$ for a given
hypercyclic operator and study the new notion of reiteratively hypercyclic
operators.
