Covering spheres and balls with smaller balls

Given a sphere or a ball of radius r > 1 in an Euclidean space of dimension n, we study their thinnest coverings with unit balls. Our goal is to design a covering with the lowest covering density, which is defined by an average number of unit balls needed to cover any point within a sphere. For growing n, we obtain a new upper bound on the covering density that has the order of (n ln n)/2, which is half the order established in the classic Rogers bound