Potential enhancement of the equidistribution theorem

The sequence $$\{h_n \alpha\}$$ is equidistributed mod $$1$$ if $$\alpha$$ is irrational and $$h_n = n$$. Is there a generalization of the equidistribution theorem with some sequence $$h_n$$ (other than the type $$h_n = \lfloor \beta + \gamma n\rfloor$$) that would still work? I am looking for a sequence $$h_n$$ such that $$h_n/n \rightarrow \infty$$ if at all possible. I need something like this in a specific context, see section 3.1. in this article.

I thought I read that a sequence satisfying $$\sqrt{h_{n+1}}-\sqrt{h_n} \rightarrow 0$$ would do (one such sequence is $$h_n$$ being the $$n$$-th prime number), but I asked the question on MSE (here) and the answer is negative in general.

Weyl proved in 1916 that a polynomial sequence with at least one irrational coefficient (besides the constant coefficient) is equidistributed modulo $$1$$. In particular, $$\alpha n^2$$ works, so $$h_n=n^2$$.