Covering half-grids with lines and planes

TL;DR

This work investigates hyperplane coverings of finite grid-like point sets in $\mathbb{R}^d$, focusing on covering every point at least $k$ times and, in many results, allowing a missing point. It develops lower bounds via a combinatorial lemma and LP duality, constructs explicit upper bounds, and analyzes both conical grids and half-grids, including generic and structured variants in low dimensions. The main findings include a universal lower bound for conical grids, $nk\left(1-\tfrac{1}{e}-O\left(\tfrac{1}{n}\right)\right)$, and a bound for $m\times n$ half-grids, $mk\left(1-e^{-\frac{n}{m}}-O\left(\frac{n}{m^2}\right)\right)$, with asymptotically sharp results in $2$ and $3$ dimensions for half-grids missing a vertex. In particular, in 2D the leading term in the missing-point setting is $\tfrac{3}{2}nk$, while in 3D it is $\tfrac{31}{18}nk$, and exact results are obtained for $k=1$, $d=2$ via the polynomial method. These results extend and refine the Alon–Füredi framework to multiplicity and to diverse grid geometries, offering precise asymptotics and constructive coverings.

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-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.

Theorems & Definitions (30)

• Definition 1: Conical grids
• Definition 2: Half-grids
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 2.1
• proof
• Corollary 2.2