Prove that $e^n\bmod 1$ is dense in $[0,1]$

I was asked to turn a comment into an answer, even though it was mainly a quote, not any work of my own. It's well known that the fractional parts $\{\theta^n\}$ are not just dense, but uniformly distributed for almost all $\theta$. The irony is, that for any individual $\theta$, we don't know nearly as much. Let's quote http://www.kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B34/pdf/B34_009.pdf

For instance, we cannot disprove that $\displaystyle \lim\{e^{n}\}=0$,where $\{x\}$ is the fractional part of a real number $x$. In the case where $\alpha$ is a transcendental number, it is generally difficult to prove that the sequence $\{\alpha^{n}\}(n=0,1, \ldots)$ has two distinct limit points.

So there's little hope concerning transcendental numbers like $e$, and the results for algebraic $\theta$ aren't exactly mind-boggling, either. We can but hope that there will be some progress, soon.