Oh that's brilliant, yes it's just Weyl's theorem for one number combined with pigeonhole principle, so simple.

Thank you, such a useful comment! @mathworker21 I wanted to ask if I followed you argument correctly. Assuming that all $x_k$ are rationally independent, we can immediately apply the generalized Weyl's criterion on $k$ irrationals to show that $\{(Lx_1, ..., Lx_n)\}_{L=1}^\infty$ is dense in $[0, 1]^n$. Now in the most general setting, we can have disjoint groups of $x_k$'s denoted by $\{S_i\}_{i=1}^m$ where $\cup_{i=1}^m S_i = [n]$ and each group $\{x_k: k \in S_i\}$ is rationally dependent. I am still not entirely sure how to apply the "coprime to common denominator" here.