Covering Spheres with Spheres

Given a sphere of any radius r in an n-dimensional Euclidean space, we study the coverings of this sphere with solid spheres of radius one. Our goal is to design a covering of the lowest covering density, which denes the average number of solid spheres covering a point in a bigger sphere. For growing dimension n, we design a covering that gives the covering density of order (n ln n)/2 for a sphere of any radius r> 1 and a complete Euclidean space. This new upper bound reduces two times the order n ln n established in the classic Rogers bound. [PUBLICATION ABSTRACT]