How to show $\{a^n \bmod \alpha\}_{n \in \mathbb{N}}$ is dense in $[0,\alpha]$ if $a > 1$ is trancendental over ${\mathbb Q}[\alpha]$

Question

How to show $\{a^n \bmod \alpha\}_{n \in \mathbb{N}}$ is dense in $[0,\alpha]$ if $a > 1$ is trancendental over ${\mathbb Q}[\alpha]$? If $a$ is transcendental over ${\mathbb Q}[\alpha]$ then the integer multiples $\{na \bmod \alpha\}_{n \in \mathbb{N}}$ are dense in $[0,\alpha]$. But what about the positive integer powers of $a$ when $a > 1$? It seems like $\{a^n \bmod \alpha\}_{n \in \mathbb{N}}$ should be dense if $a > 1$ is transcendental over ${\mathbb Q}[\alpha]$. But how to prove it?

Answer 1

This isn't true. According to the second paragraph of this paper, Vijayaraghavan has shown (in particular) the following:

There are uncountably many numbers $a>1$ such that $\{a^n\}$ is not dense in $[0,1]$.

In particular, there must be such a transcendental $a$. Hence the claim is false for $\alpha=1$.