An infinite packing theorem for spheres

Abstract

than that of the series with general term e-e2, and to wonder further what the order of magnitude of these circles might be, and whether one could determine it directly. It turned out that Borel's bound was incredibly weak: the convergence is actually so much slower that the radii of these circles can be shown to form a divergent series! That is to say, to state it simply for the unit circle, if the radii { rk } of the disjoint circles satisfy the condition >3= r.=1, then >It_rk= 00 A proof of this, using the machinery of uniform approximations to functions of a complex variable, can be found in Mergelyan [3]. In this paper we give a direct and simple proof of this geometric result. In fact, our method is such that it yields at once a result of greater generality: we shall show that the infinite packing theorem just mentioned holds not only for circles in the plane, but that an analogous result holds for spheres in n-space, as well as for more general figures.