A class of lacunary trigonometric series

It is shown that there exists a sequence of natural numbers $\{n_k\}$ which does not belong to the class $B_2$ and which cannot be decomposed into a finite number of lacunary sequences such that: a) if the series $\sum_{k=-\infty}^\infty c_ke^{in}k^x$ converges on a set of positive measure, then the series consisting of the squares of the coefficients converges; b) for each set $E$ of positive measure we can remove from the system $\{e^{in}k^x\}_{k=-\infty}^\infty$ a finite number of terms with the result that what is left is a Bessel system in $L^2(E)$; and c) if the series $\sum_{k=-\infty}^\infty c_ke^{in}k^x$ converges to zero on a set of positive measure, then each coefficient is zero.