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 (n^{d ( d - k ) / ( d - 1 ) - ε}) and at most O(n^{d ( d - k ) / ( d - 1 )}) , where ε> 0 is an arbitrarily small constant. This nearly settles a problem mentioned in the book by Brass et al. (Research problems in discrete geometry, Springer, New York, 2005). 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 (Comput Geom 25(1–2):13–20, 2003). 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^d and an arrangement of m hyperplanes in R^d with no K_{r , r} in their incidence graph and with at least Ω ((mn) ^{1 - ( 2 d + 3 ) / ( ( d + 2 ) ( d + 3 ) ) - ε}) incidences if d is odd and Ω((mn)1-(2d2+d-2)/((d+2)(d2+2d-2))-ε) incidences if d is even.