A local limit theorem for the distribution of fractional parts of an exponential function

Let $\Delta=[\alpha,\beta]\subset[0,1]$, $|\Delta|=\beta-\alpha$. Let $\chi(t)$ be the indicator function of $\Delta$. The number of fractional parts of $\{\xi2^x\}$, $x=0,1,\dots,n-1$, belonging to the segment $\Delta$ is
$$N_n(\xi,\Delta)=\sum_{x=0}^n \chi(\{\xi2^x\}).$$
Let $\tau$ be a non-negative integer number. The following result is proved:
Theorem. For $n\to\infty$, uniformly in $\tau$,
$$\operatorname{mes}\{\xi:0\le\xi\le 1,\,N_n(\xi,\Delta)=\tau\}= \frac{1}{\sigma\sqrt{2\pi n}}\exp \biggl(-\frac{(\tau-n|\Delta|)^2}{2n\sigma^2}\biggr)+O\biggl(\frac{\sqrt{\ln n}}{n}\biggr).$$