While reading other topics, e,g, Is $n \sin n$ dense on the real line? or Is $\{ \sin n^m \mid n \in \mathbb{N} \}$ dense in $[-1,1]$ for every natural number $m$?, the following problem appeared in my head:
- is $\{\sin(x^n)|n\in\mathbb{N}\}$ dense in $[-1,1]$ for all $x>1$?
or a weaker problem:
- if $x>1$, then $\lim_{n\to\infty} \sin(x^n)$ does not exist?
I proved the second one for $x=2$ and $x=3$ (with use of sine/cosine multiple angle formulas) and have some thoughts for $x\in\mathbb{N}$, but I have completely no idea how to deal e.g. with $x=e$.
The equality $x^n=e^{n\ln x}$ (for $x>1$) and Zeldich’ attraction theorem applied to a function $f(t)=e^t$, any number $0\le x_0\le 2\pi$, and any open neighborhood $A$ of the set $2\pi\Bbb Z+x_0$ imply that a set $X$ of $x>1$ for which the first problem has a negative answer, is meager. On the other hand, Proposition from the same answer implies that $|X|=2^{\aleph_0}$.