The analogue of the law of large numbers for additive functions on sparse sets

An analog of the Turan'n–Kubilyus inequality is proved for a sufficiently wide class of sequences which contains, in particular, $a_n=f(n)$ and $a_n=f(p_n)$, where $f(n)$ is a polynomial with integral coefficients. This result helps us to obtain integral limit theorems for additive functions on the class of sequences under investigation.