SCHWEITZER 2003 Problem 5

Let $d>1$ be integer and $0<r<\frac12$ . Show that there exist finitely many (depending only on $d,r$ ) nonzero vectors in $\mathbb{R}^d$ such that if the distance of a straight line in $\mathbb{R}^d$ from the integer lattice $\mathbb{Z}^d$ is at least $r$ , then this line is orthogonal to one of these finitely many vectors.

(translated by L. Erdős)