Generalizations of the Sidon-Telyakovskiĭ theorem

The well-known Sidon-Telyakovskii integrability condition is considerably lightened as follows:

$$\frac{1}{n}\sum\limits_{k = 1}^n {\frac{{\vert\Delta c(k){\vert^p}}}{{A_k^p}} = O(1),\quad n \to \infty } ,$$

where $\{ c(n)\}$ is a certain null-sequence and $1 < p \leq 2$. It is proved that $\sum\nolimits_{n = 1}^\infty {{n^{p - 1}}\vert\Delta c(n){\vert^p}{\rho ^p}(n) < \infty }$ is also a sufficient integrability condition provided $\sum\nolimits_{n = 1}^\infty {(1/n\rho (n)) < \infty }$, where $\{ \rho (n)\}$ is an increasing sequence of positive numbers.