The absolute convergence of lacunary series

A theorem is proved from which it follows that there exists a complete $U$-set $E$ and a number $p$ such that: a) if the $p$-lacunary trigonometric series
$$ \sum_{k=1}^\infty a_k\sin(n_kx+\varepsilon_k), \qquad \varliminf_{k\to\infty}n_{k+1}/n_k>p, $$
converges on $E$, the series of the moduli of its coefficients converges; b) if the sum of the $p$-lacunary trigonometric series is differentiable on $E$, it is continuously differentiable everywhere.