On coverings of four-space by spheres
F. L. Cleaver · Transactions of the American Mathematical Society · 1965
Let E" denote Euclidean n-space and S" the unit hypersphere about the origin.A lattice L in £" is defined to be the totality of all points of the form SiXy + g2X2 + ■■■ + gnXn where gy,g2,---,g" axe arbitrary integers and X1,X2,--.,XB is a fixed set of linearly independent points of £".The set Xy,X2,-,Xn is called a basis of L. The determinant of L, written d(L), is defined to be the determinant of the basis Xy,X2,---,X".Lis said to be S-admissible, for a set Sin E", if S contains no points of Lin its interior other than the origin.Define the critical determinant A(S) to be inf{|ci(L)|:Lis an S-admissible lattice}, if there exists at least one ^-admissible lattice, and co otherwise.If A(S) is finite and if there exists an S-admissible lattice L such that d(L) = A(S), then