Density space theory : discrete spaces

Graduation date: 1971Allen Freedman defined a density space to be the ordered pair\ud (S,\u1d4d5) where S is a certain kind of semigroup called an s-set and \u1d4d5\ud is a special type of family of finite subsets of S called a fundamental\ud family on S. Several properties for density spaces are\ud obtained, and the theory of a particular class of density spaces,\ud called discrete density spaces, is developed in detail. The theory of\ud discrete density spaces includes generalizations of some theorems of\ud H. Mann, F. Dyson, and Freedman. Several inequalities for the\ud density of the sum of two subsets of the s-set are obtained. Particular\ud emphasis is placed on obtaining some sufficient conditions for the\ud a + β property of additive number theory