Covering Lattice Points by Subspaces and Counting Point-Hyperplane Incidences

Let d and k be integers with 1 ≤ k ≤ d − 1. Let Λ 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 Λ ∩ K. In particular, our results imply that the minimum number of k-dimensional linear subspaces needed to cover the d-dimensional n × · · · × n grid is at least Ω(nd(d−k)/(d−1)−ε) and at most O(nd(d−k)/(d−1)), where ε > 0 is an arbitrarily small constant. This nearly settles a problem mentioned in the book of Brass, Moser, and Pach [7]. We also find tight bounds for the minimum number of k-dimensional affine subspaces needed to cover Λ ∩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 [6]. For d ≥ 3 and ε ∈ (0, 1), we show that there is an integer r = r(d, ε) such that for all positive integers n,m the following statement is true. There is a set of n points in R and an arrangement of m hyperplanes in R with no Kr,r in their incidence graph and with at least Ω ( (mn)1−(2d+3)/((d+2)(d+3))−ε ) incidences if d is odd and Ω ( (mn)1−(2d+d−2)/((d+2)(d+2d−2))−ε ) incidences if d is even. 1998 ACM Subject Classification G.2.1 Combinatorics