Is the sequence $\{\pi^n\}=\pi^n-\lfloor\pi^n\rfloor$ dense? In other words for any given $\varepsilon>0$ and $t\in[0,1]$ is there a proper $n\in\mathbb{N}$ satisfying $|\{\pi^n\}-t|<\varepsilon$ ?
(*) What is the condition on $q$ to make the sequence $\{q^n\}$ dense?
I know the necessary and sufficient condition for $\{nq\}$ is $q\not\in\mathbb{Q}$.
Also, to make the question * nicer, extend it for all (positive and negative) integers.
For $(\ast)$ see Power Fractional Parts and the references there. Hardy and Littlewood (1914) proved that the sequence of fractional parts $q^n-\lfloor q^n\rfloor$ is equidistributed for almost all real numbers $q>1$. One exceptional number is the golden ratio. I don't know if $\pi$ is another exception, probably not. I did not find a specific result about $\pi$. I think that it is an open problem.