Let $a \in \mathbb{R}\backslash\mathbb{Q}$ be an generic irrational number. Then, it's well-known that $\{ n a \text{ mod } 1\mid n\in \mathbb{N}\}$ is equidistributed on [0, 1).
My question is, choose any $\theta \in [0, 1)$ and consider the subset $G_{\theta, \delta} := \{ n \mid n \in \mathbb{N} \text { and } |(n a \text{ mod } 1) - \theta|<\delta\}$ for some $\delta > 0$, and see what happens on $\{ n^k a \text{ mod } 1 \mid n \in G_{\theta, \delta}\}$ for generic $\theta$ and for all positive $\delta$. In here $k$ is any positive integer larger than $1$.
I guess it's also equidistributed on [0, 1) or at least, it's not small/local as $G_{\theta, \delta}$ does near $\theta$, and so I've tried to use Weyl's criterion. However, it's a little complicated to simplify.
In this question, I use "local" as the unions of little numbers of small neighborhood of some points.
Any help or comments would be appreciated.