Covering space with convex bodies

1. A few years ago Rogers [1] showed that, if K is any convex body in n-dimensional Euclidian space, there is a covering of the whole space by translates of K with density less than nlogn+nloglogn+5n, provided n > 3. However the fact that the covering density is reasonably small does not imply that the maximum multiplicity is also small. In the natural covering of space by closed cubes, the density is 1, but each cube vertex is covered 2 n times. Our object in this note is to prove that, provided n is sufficiently large, there is, for each convex body K, a covering with density less than nlogn=, nloglogn ;-4n , and such that no point is covered more than e {n log n + nloglogn + 4n} times. By dimension theory, some points must be covered n + 1 times. 2. In this section we take K to be a Lebesgue measurable set with finite positive measure V. Further let A be the lattice of all points with integral coordinates, and suppose all the distinct translates of K by the vectors of A are disjoint. We suppose that N points x1 , x,,. .. , xN are chosen at random in the cube C of points x with 0<xl <1, 0<x2 <1,. . ., 0<xfl <1, and investigate the average density of the set of points covered exactly k times by the system of sets (1) K+xi +g