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

D. A. Moskvin   ; A. G. Postnikov

Teoriâ veroâtnostej i ee primeneniâ, Tome 23 (1978) no. 3, pp. 540-547

Résumé

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 $\tau$, $$ \operatorname{mes}\{\xi:0\le\xi\le 1,\,N_n(\xi,\Delta)=\tau\}= \frac{1}{\sigma\sqrt{2\pi n}}\exp \biggl(-\frac{(\tau-n|\Delta|)^2}{2n\sigma^2}\biggr)+O\biggl(\frac{\sqrt{\ln n}}{n}\biggr). $$