TITLE: The fractional parts of the powers of the golden ratio are not equidistributed in [0,1]
QUESTION [6 upvotes]: Let $$a_n=\left(\frac{1+\sqrt{5}}{2}\right)^n.$$
For a real number $r$, denote by $\langle r\rangle$ the fractional part of $r$. 
Why is the sequence $$\langle a_n\rangle$$ not equidistributed in $[0,1]$?

REPLY [17 votes]: Let $\phi'=(1-\sqrt 5)/2$ denote the Galois conjugate of the golden mean $\phi$. 
Then $\phi^n+\phi'^{n}$ is an integer for every $n\in\mathbb N$, i.e.
$$\phi^n+\phi'^{n}\equiv 0\ (\mbox{mod}\ 1).$$
But $|\phi'|<1$, so $\phi'^{n}\to 0$. This implies that $\phi^n\to 0\ (\mbox{mod}\ 1)$.
The property that the sequence  $\{ \mbox{frac} (x^n)\}$ is not equidistributed is shared by other Pisot numbers. There is quite a lot of research publications devoted to them.