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Density functions with respect to a model growth function

Year: 2025
Type: article
Abstract: 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 $L(r)=M^{-\rho}(r)V(r))$ has a slow growth with respect to $M$. The concept of $\rho$-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 $N_M(\alpha)$ and lower density $\underline N_M(\alpha)$ of a function $f$ is obtained. A uniformity theorem is proved. The main properties of $\rho$-additive and $\rho$-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.
DOI: 10.4213/sm10188e
Language: English
Open Access status: bronze
OpenAlex ID: W4415360804
FWCI: 0.603
Cites: 12
Cited by: 1
Related to: 10