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.