Covering a sphere with caps: Rogers bound revisited

We consider coverings of a sphere Sn r of radius r with the balls of radius one in an n-dimensional Euclidean space R. Our goal is to minimize the covering density, which defines the average number of the balls covering a point in Sn r . For a growing dimension n, we obtain the covering density at most (n lnn)/2 for any sphere Sn r and the entire space R. This new upper bound reduces two times the density n lnn established in the classical Rogers bound.