Intermediate-scale statistics for real-valued lacunary sequences

We study intermediate-scale statistics for the fractional parts of the sequence $\left(\alpha a_{n}\right)_{n=1}^{\infty}$ , where $\left(a_{n}\right)_{n=1}^{\infty}$ is a positive, real-valued lacunary sequence, and $\alpha\in\mathbb{R}$ . In particular, we consider the number of elements $S_{N}\!\left(L,\alpha\right)$ in a random interval of length $L/N$ , where $L=O\!\left(N^{1-\epsilon}\right)$ , and show that its variance (the number variance) is asymptotic to L with high probability w.r.t. $\alpha$ , 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 $\alpha\in\mathbb{R}$ when $L=O\!\left(N^{1/2-\epsilon}\right)$ . For slowly growing L, we further prove a central limit theorem for $S_{N}\!\left(L,\alpha\right)$ which holds for almost all $\alpha\in\mathbb{R}$ .