A GENERAL THEORY OF DENSITY IN ADDITIVE NUMBER THEORY

Abstract approved (Major pr fessor) L. Schnirelmann's definition of density for sets of positive integers is generalized to density for subsets of certain semigroups called s-sets. Let S be an s-set. For a subset X of S and a finite subset D of S, let X(D) denote the number of elements in the set X (---D. Let 2:1 be any family or non-empty finite sub-sets of S. Then the density of a subset A of 5, with respect gib fA(G)IGETtil. 1S(G) I Axioms are presented which define a family of finite sub-sets of S to be a fundamental family on S. We require in the above definition that ki be either a fundamental family or a cer-tain subfamily of a fundamental family called the family of all Cheo sets of the fundamental family. The two densities obtained in this wa