On the order of growth $o(\log\log n)$ of the partial sums of Fourier–Stieltjes series of random measures

Random measures of the form
$$ \sum_{i=1}^\infty m_i\delta_{\theta_i}, \qquad \sum_{i=1}^\infty|m_i|<\infty, $$
are considered, where $\delta_{\theta_i}$ is a unit mass concentrated at the point $\theta_i\in(0;2\pi)$. For any sequence of natural numbers $\{l_k\}_{k=1}^\infty$ it is established that for almost all sequences $\theta=\{\theta_i\}_{i=1}^\infty$ the partial sums $S_{l_k}(x;d\mu_\theta)$ of the Fourier–Stieltjes series of the measure have order $o(\log\log k)$ for almost all $x\in(0;2\pi)$. As proved by Kahane in 1961, the order $o(\log\log k)$ cannot be improved. This result is connected with the well-known problem of Zygmund of finding the exact order of growth of the partial sums of Fourier series almost everywhere.