For a fixed positive integer n,
let Q be the set of all ndimensional lattice points (x_{1},⋯,x_{n}) with each x_{i} a
nonnegative integer and at least one x_{i} positive. A finite nonempty subset
R of Q is called a fundamental set if for every (r_{1},⋯,r_{n}) in R, all vectors
(x_{1},⋯,x_{n}) of Q with x_{i} ≦ r_{i}, i = 1,⋯,n, are also in R. If A is any subset of Q
and R is any fundamental set, let A(R) denote the number of vectors in
A ∩ R. Finally, if A is any proper subset of Q, let the density of A be the
quantity
taken over all fundamental sets R for which A(R) < Q(R). Then the theorem proved
in this paper can be stated as follows.
Theorem. Let A and B be subsets of Q, let C be the set of all vectors of the form
a, b, or a + b where a ∈ A and b ∈ B, let α be the density of A, and let R be any
fundamental set such that (1) there exists at least one vector in R which is not in C,
and (2) for each b in B ∩R (if any) there exists g in R but not in C such that g −b is
in Q. Then
