Density functions

A. F. Grishin ; T. I. Malyutina

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 7 (2000), pp. 387-414.

Résumé

We study properties of $\rho$-semiadditive functions, $N(\alpha +\beta )\le N(\alpha )+(1+\alpha {)}^{\rho }N\left(\frac{\beta }{1+\alpha }\right)$. Their theory is similiar to the very investigated one of semiadditive functions. The functions of density $N(\alpha )$=$lim sup{r}^{-\rho }(f(r+\alpha r)-f(r))$ $(r\to \mathrm{\infty })$ are $\rho$-semiadditive. One of results of the note is an extension of the theorem of Polya (1929) on existence of maximal and minimal densities. We are interested in the question of uniformity in the above limiting relation.