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

<jats:p>We consider families of lines that cover every point of theinteger lattice Z2, subject to the constraint that no two lines of differentdirection in the family meet at a lattice point. Restricting to latticelines (lines containing at least two, hence infinitely many, lattice points,equivalently of rational direction), we show that the set of directionsoccurring in such a covering can be made dense in the space of linedirections. The construction is a recursive splitting of Z2 into nestedrank-2 sublattice cosets, each handed off to a freshly chosen direction;the key technical point is a steering lemma showing that at every stageof the recursion a new direction arbitrarily close to any prescribed targetcan still be realized, via an elementary sieve bound.</jats:p>