AN INDEXED SET OF DENSITY BOUNDS ON LATTICE PACKINGS

It has long been known that the admissibility of a lattice Gamma with respect to a symmetric convex body B is equivalent to Gamma being a packing lattice for 1/2 B. This fact is the basis of the interplay between the classical theory of the arithmetic minima of positive definite quadratic forms, on the one hand, and the dense lattice packing of spheres in R(n), on the other. We give an indexed set of bounds delta(L)(B) greater than or equal to aj, where 0 less than or equal to j less than or equal to n/2, on the lattice packing density of B. The case j = 0 reduces to the aforementioned long-known fact, and j = 1 was proved by Elkies, Odlyzko, and Rush, and was used to obtain record high packing densities for various superballs. The new cases make possible the use of smaller primes in the construction of these dense packings.