Covering lattice points by subspaces and counting point-hyperplane incidences

Abstract

Let $d$ and $k$ be integers with $1 \leq k \leq d-1$. Let $\Lambda$ be a $d$-dimensional lattice and let $K$ be a $d$-dimensional compact convex body symmetric about the origin. We provide estimates for the minimum number of $k$-dimensional linear subspaces needed to cover all points in $\Lambda \cap K$. In particular, our results imply that the minimum number of $k$-dimensional linear subspaces needed to cover the $d$-dimensional $n \times \cdots \times n$ grid is at least $\Omega(n^{d(d-k)/(d-1)-\varepsilon})$ and at most $O(n^{d(d-k)/(d-1)})$, where $\varepsilon>0$ is an arbitrarily small constant. This nearly settles a problem mentioned in the book of Brass, Moser, and Pach. We also find tight bounds for the minimum number of $k$-dimensional affine subspaces needed to cover $\Lambda \cap K$. We use these new results to improve the best known lower bound for the maximum number of point-hyperplane incidences by Brass and Knauer. For $d \geq 3$ and $\varepsilon \in (0,1)$, we show that there is an integer $r=r(d,\varepsilon)$ such that for all positive integers $n,m$ the following statement is true. There is a set of $n$ points in $\mathbb{R}^d$ and an arrangement of $m$ hyperplanes in $\mathbb{R}^d$ with no $K_{r,r}$ in their incidence graph and with at least $\Omega\left((mn)^{1-(2d+3)/((d+2)(d+3)) - \varepsilon}\right)$ incidences if $d$ is odd and $\Omega\left((mn)^{1-(2d^2+d-2)/((d+2)(d^2+2d-2)) -\varepsilon}\right)$ incidences if $d$ is even.