Intermediate-scale statistics for real-valued lacunary sequences

We study intermediate-scale statistics for the fractional parts of the sequence, where is a positive, real-valued lacunary sequence, and. In particular, we consider the number of elements in a random interval of length, where, and show that its variance (the number variance) is asymptotic to L with high probability w.r.t., which is in agreement with the statistics of uniform i.i.d. random points in the unit interval. In addition, we show that the same asymptotic holds almost surely in when. For slowly growing L, we further prove a central limit theorem for which holds for almost all.