Let $x\in \mathbb{R}^n$. Are there always sequences $z_k\in \mathbb{Z}^n$ and $t_k\in\mathbb{Z}$ such that $|z_k-t_kx|\rightarrow 0$ when $k\rightarrow \infty$? (The interesting case is when $x\in \mathbb{R}^n\setminus\mathbb{Q}^n$).
In the $1$-dimensional case, I believe this is a consequence of the pidgeon hole principle. The application of this has to do with positive definite quadratic forms.
I found the answer to be yes! Using Kronecker's theorem https://en.wikipedia.org/wiki/Kronecker%27s_theorem, and letting $\beta_j=0$ for all $j$ and $m=1$ the result follows.