On divergent lacunary trigonometric series

Abstract:

Let $S = \left \{ {{\lambda _n}} \right \}_{n = 1}^\infty$ be a sequence of positive real numbers such that ${\lambda _{n + 1}}/{\lambda _n} \geqslant q > 1$ for all $n$. If $q > 8$ then any divergent series with frequencies in $S$ has its real part diverging (uniformly) to $+ \infty$ on a set of positive logarithmic capacity. It is necessary that $q > 2$. A new sufficient condition for the generalized capacity of a set to be positive is developed and then applied in the proof.