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

I just noticed that I have left unanswered one part of an old multi-part question and so decided to re-ask it separately:

Consider the sequence $$e^n\bmod 1$$, $$n\in\Bbb N$$. Show that it is dense in $$[0,1]$$.

This apparently does require specific (approximation?) properties of $$e$$, as for example replacing $$e$$ with any integer leads to a non-dense sequence. On the other hand, for every sequence of numbers $$a_n\in(0,1)$$, it is not hard to find $$\alpha$$ such that $$|\alpha^{2^n}\bmod 1- a_n|<\frac1n$$ for all $$n$$, or $$\beta$$ such that $$|\beta^n\bmod 1-a_n|<\frac1{1000}$$. Hence there exist (irrational) bases that lead to a dense sequence and others that lead to a non-dense sequence. Other than that I'm a bit at a dead end.

• I take it you mean $e^n\mod 1$ ? – Olivier Bégassat Nov 18 '20 at 16:03
• I think $$\text{frac}\left(e^n\right)$$ is more expressive – Raffaele Nov 18 '20 at 16:22
• $x_n \operatorname{mod1} =\big\{ x_n \big\}$ where $\big\{ x_n \big\}= x_n -[x_n].$ – Young-san Nov 18 '20 at 17:38
• "For the exponential this is a bit more difficult." In fact, it's a bit too difficult for mathematics as we know it, today. – user436658 Nov 18 '20 at 17:55
• kurims.kyoto-u.ac.jp/~kenkyubu/bessatsu/open/B34/pdf/… – user436658 Nov 18 '20 at 22:11

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.