Density functions with respect to a model growth function

The properties of general density functions with respect to a model growth function M and related semiadditive functions are discussed. The concept of function of slow growth with respect to the model growth function M is introduced; it is shown that the function L(r)=M-ρ(r)V(r)) has a slow growth with respect to M. The concept of ρ-semiadditive function with respect to M is also introduced, and the main properties of such functions are established. Density functions are studied; a criterion of the continuity of the density NM(α) and lower density NM(α) of a function f is obtained. A uniformity theorem is proved. The main properties of ρ-additive and ρ-semiadditive functions with respect to the model function M are presented. One of the central results is a theorem that can be viewed as an extension of Pólya's theorem on the existence of minimal and maximal densities to a wider class of functions, whose growth is bounded by an arbitrary model growth function M. Examples of functions f and their density functions are presented. Bibliography: 17 titles.