We study hyperplane covering problems for finite grid-like structures in $\mathbb{R}^d$. We call a set $\mathcal{C}$ of points in $\mathbb{R}^2$ a conical grid if the line $y = a_i$ intersects $\mathcal{C}$ in exactly $i$ points, for some $a_1 > \cdots > a_n \in \mathbb{R}$. We prove that the number of lines required to cover every point of such a grid at least $k$ times is at least $nk\left(1-\frac{1}{e}-O(\frac{1}{n}) \right)$. If the grid $\mathcal{C}$ is obtained by cutting an $m \times n$ grid of points in half along one of the diagonals, then we prove the lower bound of $mk\left(1-e^{-\frac{n}{m}}-O(\frac{n}{m^2})\right)$. In general, we call a grid obtained by cutting a grid in $\mathbb{R}^d$ along one of the diagonals a half-grid. Motivated by the Alon-Füredi theorem on hyperplane coverings of grids that miss a point and its multiplicity variations, we study the problem of finding the minimum number of hyperplanes required to cover every point of an $n \times \cdots \times n$ half-grid in $\mathbb{R}^d$ at least $k$ times while missing a point $P$. For almost all such half-grids, with $P$ being the corner point, we prove asymptotically sharp upper and lower bounds for the covering number in dimensions $2$ and $3$. For $k = 1$, $d = 2$, and an arbitrary $P$, we determine this number exactly by using the polynomial method bound for grids.

翻译：我们研究 $\mathbb{R}^d$ 中有限网格状结构上的超平面覆盖问题。若存在 $a_1 > \cdots > a_n \in \mathbb{R}$ 使得直线 $y = a_i$ 与 $\mathbb{R}^2$ 中的点集 $\mathcal{C}$ 恰好相交于 $i$ 个点，则称 $\mathcal{C}$ 为锥形网格。我们证明，覆盖此类网格中每个点至少 $k$ 次所需的直线数至少为 $nk\left(1-\frac{1}{e}-O(\frac{1}{n}) \right)$。若网格 $\mathcal{C}$ 是通过沿着对角线将 $m \times n$ 点网格的一半切割得到，则我们证明下界为 $mk\left(1-e^{-\frac{n}{m}}-O(\frac{n}{m^2})\right)$。一般而言，我们将通过沿对角线切割 $\mathbb{R}^d$ 中的网格得到的网格称为半格子。受关于缺失一点及其多重性变体的网格超平面覆盖的 Alon-Füredi 定理的启发，我们研究在 $\mathbb{R}^d$ 中 $n \times \cdots \times n$ 半格子上，覆盖每个点至少 $k$ 次同时缺失一点 $P$ 所需的最少超平面数问题。对于几乎所有此类半格子（其中 $P$ 为角点），我们证明了在维度 $2$ 和 $3$ 下覆盖数的渐近尖锐上下界。当 $k = 1$、$d = 2$ 且 $P$ 任意时，我们利用网格的多项式方法界精确确定了该数值。