Let $$(a_n)_{n \in \mathbb{N}} = (1,2,3,4,6,8,9,12,16,18,\cdots)$$ be a sequence that is a listing of the set $$\{2^n3^m \mid m,n \in \mathbb{N}\}$$ We need to find an irrational number $\alpha$ such that $(a_n)\alpha\pmod1$ is not equidistributed.

I have a rough idea about it. I think that it may be related to normal number base $2$ and normal number base $3,$ because we know that let $x \in [0,1]$, then $x$ is normal in base $2$ if and only if $2^n x \bmod 1$ is equidistributed. And same arguments work for numbers normal in base $3$. But I'm not sure how to proceed from here. Could anyone help me?