On the directions occurring in lattice-line coverings of the integer plane

We consider families of lines that cover every point of the integer lattice $\mathbf{Z}^2$ in the plane, subject to the constraint that no two lines of different direction in the family meet at a lattice point. Restricting to \emph{lattice lines} (lines containing at least two, hence infinitely many, lattice points, equivalently of rational direction), we show that the set of directions occurring in such a covering can be made dense in the space of line directions. The construction is a recursive splitting of $\mathbf{Z}^2$ into nested rank-2 sublattice cosets, each handed off to a freshly chosen direction; the key technical point is a steering lemma showing that at every stage of the recursion a new direction arbitrarily close to any prescribed target can still be realized, via an elementary sieve bound.