The fractional parts of the powers of the golden ratio are not equidistributed in [0,1]

Solution 1:

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.

Solution 2:

If you just try it, the fractional part becomes very close to 0 or 1 quickly. This is because $\phi ^2=\phi +1$.