Random Packings and Coverings of the Unit n-Sphere

Abstract

It is well known that the quantity M <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</inf> (n, θ), the maximum number of nonoverlapping spherical caps of half angle θ (a “packing”) which can be placed on the surface of a unit sphere in Euclidean n-space is not less than exp [– n log sin 2θ + o(n)] (θ < π/4). In this paper we give a new proof of this fact by a “random coding” argument, the central part of which is a theorem which asserts that if a set of roughly exp (–n log sin 2 θ) caps is chosen at random, that on the average only a very small fraction of the caps will overlap (when n is large). A related problem is the determination of M <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</inf> (n, θ), the minimum number of caps of half angle θ required to cover the unit Euclidean n-sphere. We show that M <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">c</inf> (n, θ) = exp [–n log sin θ + o(n)]. The central part of the proof is also a random coding argument which asserts that if a set roughly exp (–n log sin θ) caps is chosen at random, that on the average only a very small fraction of the surface of the n-sphere will remain uncovered (when n is large).