Consider the lattice $\mathbb{Z}^{n}\subset\mathbb{R}^{n}$ and a line (not necessarily through the origin).
What conditions can be placed on the slope of the line that is necessary and sufficient so that the Euclidean distance from lattice points to the line are arbitrarily close to $0$?
Note that the lines does not necessarily pass through the origin, otherwise Minkowski's theorem would have solved it.
Thank you.