On the connection between the Beurling-Malliavin density and the asymptotic density

We study the notion of Beurling-Malliavin density from the point of view of Number Theory. We prove a general relation between the Beurling-Malliavin density and the upper asymptotic density; we identify a class of sequences for which the two densities coincide; this class contains the arithmetic progressions. Last, by means of an alternative definition of Beurling-Malliavin density, we study the connection with the asymptotic density for another kind of sequences that again generalizes the arithmentic progressions.