On the existence of universal series by trigonometric system

In this paper we prove the following: let $\omega(t)$ be a continuous function, increasing in $[0,\infty)$ and $\omega(+0)=0$. Then there exists a series of the form$\sum_{k=-\infty}^\infty C_ke^{ikx}$ with $\sum_{k=-\infty}^\infty C^2_k \omega(|C_k|)<\infty$, $C_{-k}=\bar{C}_k$, with the following property: for each $\epsilon>0$ a weighted function $\mu(x), 0<\mu(x) \le1,| \{x\in[0,2\pi]: \mu(x)\not =1 \}| <\epsilon$ can be constructed, so that the series is universal in the weighted space $L_\mu^1[0,2\pi]$ with respect to rearrangements.