AI 中文总结
该研究针对整数平面格线覆盖，通过递归拆分秩2子格陪集的构造，证明此类覆盖的方向集合可在直线方向空间中稠密。
AI 中文摘要
我们研究覆盖平面上整数格点$\boldsymbol{Z}^2$每一点的直线族，约束条件为该族中不同方向的任意两条直线不会交于格点。我们将范围限定在「格线」（即包含至少两个、因而无穷多个格点的直线，等价于有理方向的直线），证明此类覆盖中出现的方向集合可在直线方向空间中实现稠密性。该构造是将$\boldsymbol{Z}^2$递归拆分为嵌套的秩2子格陪集，每个陪集分配一个新选定的方向；关键技术点是一个导向引理，表明在递归的每一步，仍可通过基础筛法界实现任意接近指定目标的新方向。
英文摘要
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.
Comments21 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
