Numbers whose powers are almost integers

Some real numbers $\alpha$ have the property that their powers get ever closer to being integers -- more precisely, that $$ \lim_{n\to\infty} \alpha^n-[\alpha^n] = 0 $$ where $[\cdot]$ is the round-to-nearest-integer function.

This is trivially the case when $\alpha$ is itself an integer as well as when $|\alpha|<1$. But there are also other numbers with this property, such as $\frac{1+\sqrt 5}{2}$ (the golden ratio), $2+\sqrt3$, or $\frac{5+\sqrt{13}}2$. (The trick for each of these is that $\alpha^n+\beta^n$ solves a second-order integer recurrence, where $|\beta|<1$).

Just to be sure this is not trivial, there numbers without this property, such as $\sqrt k$ for any nonsquare integer $k$.

Is there a name for this property? Or a general theory of such numbers? Are there nontrivial ones that are not quadratic over $\mathbb Q$?

they are called Pisot numbers, after a 1938 thesis, though Thue in 1912 and Hardy in 1919 also noticed them. Pisot characterized them in a rather beautiful theorem. here's a wikipedia article https://en.wikipedia.org/wiki/Pisot%E2%80%93Vijayaraghavan_number