Covering half-grids with lines and planes

TL;DR

This paper investigates hyperplane covering problems for finite grid-like structures in Euclidean space, establishing lower bounds and exact results for covering points with lines and planes, especially for half-grids and grids missing a point.

Contribution

It introduces new lower bounds for covering grid structures with hyperplanes and determines exact covering numbers for specific cases using polynomial methods.

Findings

Lower bounds for covering half-grids with lines and planes.

Exact covering number for 2D grids missing a point.

Asymptotically sharp bounds for half-grids in 2D and 3D.

Abstract

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 into a half along one of the diagonals, then we prove the lower bound of $mk\left(1-e^{-\frac{n}{m}}-O(\frac{n}{m^2})\right)$. Motivated by the Alon-F\"uredi 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.