Lattice coverings with four-dimensional spheres

Source abstract

1. Let J n be a sphere with volume V ( J n ) in the n -dimensional Euclidean space R n . Let Λ be a lattice of determinant d (Λ) such that every point in R n lies in one at least of the bodies obtained from J n by applying to it all possible lattice translations. Then Λ is called a covering lattice for J n and V ( J n )/ d (Λ) is called the density of the lattice covering by J n provided by Λ. The lower bound θ n of V ( J n )/ d (Λ) taken over all covering lattices Λ for J n is called the density of the thinnest lattice covering by J n .