An asymptotic property of gap series (Q1397473)

Consider sums \(\sum f(n_kx)\), where \(f\) is a real-valued, periodic function on \(\mathbb R\) and \(\{n_k\}\) is a strictly increasing sequence of positive integers. Under a certain gap condition, the author extends Takahashi's law of the iterated logarithm-type result for the above sums and shows that the obtained upper bound is optimal in a certain sense.