Discrete & Computational Geometry

, Volume 61, Issue 2, pp 325–354 | Cite as

Covering Lattice Points by Subspaces and Counting Point–Hyperplane Incidences

  • Martin BalkoEmail author
  • Josef Cibulka
  • Pavel Valtr


Let d and k be integers with \(1 \le k \le d-1\). 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 \(\Omega \bigl (n^{d(d-k)/(d-1)-\varepsilon }\bigr )\) and at most \(O\bigl (n^{d(d-k)/(d-1)}\bigr )\), where \(\varepsilon >0\) 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 \(\Lambda \cap K\). We use these new results to improve the best known lower bound for the maximum number of point–hyperplane incidences by Brass and Knauer (Comput Geom 25(1–2):13–20, 2003). For \(d \ge 3\) and \(\varepsilon \in (0,1)\), we show that there is an integer \(r=r(d,\varepsilon )\) such that for all positive integers nm 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 \(\Omega \bigl ((mn)^{1-(2d+3)/((d+2)(d+3)) - \varepsilon }\bigr )\) incidences if d is odd and \(\Omega \bigl ((mn)^{1-(2d^2+d-2)/((d+2)(d^2+2d-2)) -\varepsilon }\bigr )\) incidences if d is even.


Lattice point Covering Linear subspace Point–hyperplane incidence 

Mathematics Subject Classification

52C07 11H31 52C10 


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Authors and Affiliations

  1. 1.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles UniversityPraha 1Czech Republic
  2. 2.Alfréd Rényi Institute of MathematicsHungarian Academy of SciencesBudapestHungary

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