  We study intermediate-scale statistics for the fractional parts of the
sequence $(\alpha a_n)_{n=1}^{\infty}$, where $(a_n)_{n=1}^{\infty}$ is a
positive, real-valued lacunary sequence, and $\alpha\in\mathbb{R}$. In
particular, we consider the number of elements $S_{N}(L,\alpha)$ 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
asymptotics 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 $S_{N}(L,\alpha)$ which holds for almost all
$\alpha\in\mathbb{R}$.
