Abstract.Let S= {À"),^=| be a sequence of positive real numbers such that A/i+ i/An > <7 > I for all n. ff q > 8 then any divergent series with frequencies in S has its real part diverging (uniformly) to +00 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.