Dominating sets of frequencies in spectrums of measures with finite energy

A subset $\Lambda$ of $\mathbb Z$ is called a dominating set if every measure $\mu$, satisfying $\sum_{n\in\Lambda\setminus\{0\}}(|\hat\mu(n)|^2)(|n|)<+\infty$, has a finite energy $\varepsilon(\mu)=\sum_{n\in\mathbb Z\setminus\{0\}}(|\hat\mu(n)|^2)(|n|)<+\infty$. It is proved that a low density of a dominating set is positive and for every $\varepsilon>0$ there is a dominating set $\Lambda$, $\Lambda\subset\mathbb Z_+$, whose density is smaller than $\varepsilon$.