Covering triangular grids with multiplicity

TL;DR

This paper investigates the minimum number of affine hyperplanes needed to cover each point of a triangular grid multiple times, providing exact solutions for small cases and asymptotic formulas for higher dimensions.

Contribution

It introduces a new problem inspired by classical work, solves it exactly for small parameters in two dimensions, and derives asymptotic formulas for higher dimensions.

Findings

Exact solutions for $k \,\leq\, 4$ in 2D

Partial solutions for $k > 4$ in 2D

Asymptotic formulas for all $d \geq k - 2$

Abstract

Motivated by classical work of Alon and F\"uredi, we introduce and address the following problem: determine the minimum number of affine hyperplanes in $\mathbb{R}^d$ needed to cover every point of the triangular grid $T_d(n) := \{(x_1,\dots,x_d)\in\mathbb{Z}_{\ge 0}^d\mid x_1+\dots+x_d\le n-1\}$ at least $k$ times. For $d = 2$, we solve the problem exactly for $k \leq 4$, and obtain a partial solution for $k > 4$. We also obtain an asymptotic formula (in $n$) for all $d \geq k - 2$. The proofs rely on combinatorial arguments and linear programming.