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

Abstract

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.

21 pages, 10 figures. Lean formalization of all results (due to llm agent) included in anc directory, and also available at https://gitlab.liu.se/jansn19/lattice-line-covers