On Bounds for Packings on a Sphere and in Space

A method is proposed for obtaining bounds for packings in metric spaces, the method being based on the use of zonal spherical functions associated with a motion group of the space. For the maximum number $M(n,\Theta)$ of points of a unit sphere of $n$-dimensional Euclidean space at an angular distance of not less than $\Theta$ from one another, the method is used to obtain an upper bound that is better than the available ones for any fixed $\Theta\,(0<\Theta<\pi/2)$ and $n\to\infty$ This bound yields a new asymptotic upper bound for dn, namely, the maximum packing density of an $n$-dimensional Euclidean space by equal balls.