On the distribution of rational functions on consecutive powers

We show that for a prime $p$ and any nontrivial rational function $r(X) \in \mathbb{F}_p(X)$ over the finite field $\mathbb{F}_p$ of $p$ elements, the fractional parts $$ \{ \frac{r(x)}{p}, \ldots, \frac{r(x^m)}{p} \},$$ where $x$ runs through the fields elements which are not the poles of the above functions, are asymptotically uniformly distributed in the $m$-dimensional unit cube for any fixed $m$ and $p \to \infty$.