On the distribution functions of two oscillating sequences

We investigate the set of all distribution functions of two special sequences on the unit interval, which involve logarithmic and trigonometric terms. We completely characterize the set of all distribution functions G(xn) for (xn)n≥1 = ({cos(αn)n})n≥1 and arbitrary α, where {x} denotes the fractional part of x. Furthermore we give a sufficient number-theoretic condition on α for
which (xn)n≥1 = ({log(n) cos(αn)})n≥1 is uniformly distributed. Finally we calculate G(xn) in the case when α 2π ∈ Q.