A limit involving the integer nearest to $n$-th power Find all $x\in\mathbb{R}$ such that $$\lim_{n\to\infty}|x^n-\langle x^n\rangle|=0$$ where $\langle t\rangle$ is the integer nearest to $t$ (eg. $\langle\frac{1}{3}\rangle=0$, $\langle\frac{8}{3}\rangle=3$, $\langle k+\frac{1}{2}\rangle$ is not defined for $k\in\mathbb{Z}$).

I found this somewhere in the internet (today I searched again in IMO shortlists and didn't found, so it's probably not from there), tried to solve for a long time, but without nontrivial results (numbers $x\in\mathbb{Z}$ and $x\in(0,1)$ satisfy this, but I have no idea how to examine e.g. $x=\sqrt{2}$).

A Pisot number will certainly work. For instance, $\sqrt{2}+1$.
