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Covering Lattice Points by Subspaces and Counting Point–Hyperplane Incidences

  • Martin Balko
  • Josef Cibulka
  • Pavel Valtr
Article

Abstract

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.

Keywords

Lattice point Covering Linear subspace Point–hyperplane incidence 

Mathematics Subject Classification

52C07 11H31 52C10 

References

  1. 1.
    Ackerman, E.: On topological graphs with at most four crossings per edge. Submitted, preliminary version (2015) arxiv:1509.01932
  2. 2.
    Apfelbaum, R., Sharir, M.: Large complete bipartite subgraphs in incidence graphs of points and hyperplanes. SIAM J. Discrete Math. 21(3), 707–725 (2007)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Banaszczyk, W.: New bounds in some transference theorems in the geometry of numbers. Math. Ann. 296(4), 625–635 (1993)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Bárány, I., Harcos, G., Pach, J., Tardos, G.: Covering lattice points by subspaces. Period. Math. Hung. 43(1–2), 93–103 (2001)MathSciNetMATHGoogle Scholar
  5. 5.
    Brass, P., Knauer, C.: On counting point–hyperplane incidences. Comput. Geom. 25(1–2), 13–20 (2003)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Brass, P., Moser, W., Pach, J.: Research Problems in Discrete Geometry. Springer, New York (2005)MATHGoogle Scholar
  7. 7.
    Chazelle, B.: Cutting hyperplanes for divide-and-conquer. Discrete Comput. Geom. 9(2), 145–158 (1993)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Erdős, P.: On sets of distances of \(n\) points. Am. Math. Mon. 53(5), 248–250 (1946)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Erickson, J.: New lower bounds for Hopcroft’s problem. Discrete Comput. Geom. 16(4), 389–418 (1996)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Fox, J., Pach, J., Sheffer, A., Suk, A., Zahl, J.: A semi-algebraic version of Zarankiewicz’s problem. J. Eur. Math. Soc. (JEMS) 19(6), 1785–1810 (2017)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Hagerup, T., Rüb, C.: A guided tour of Chernoff bounds. Inf. Process. Lett. 33(6), 305–308 (1990)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Henk, M.: Successive minima and lattice points. Rend. Circ. Mat. Palermo (2) Suppl 70(I), 377–384 (2002)MathSciNetMATHGoogle Scholar
  13. 13.
    John, F.: Extremum problems with inequalities as subsidiary conditions. In: Studies and Essays. Presented to R. Courant on his 60th Birthday, January 8, 1948, pp. 187–204. Interscience, New York (1948)Google Scholar
  14. 14.
    Lefmann, H.: Extensions of the No-Three-In-Line problem. Submitted, preliminary version (2012). www.tu-chemnitz.de/informatik/ThIS/downloads/publications/lefmann_no_three_submitted.pdf
  15. 15.
    Mahler, K.: Ein Übertragungsprinzip für konvexe Körper. Čas. Mat. Fys. 68, 93–102 (1939)MATHGoogle Scholar
  16. 16.
    Matoušek, J.: Lectures on Discrete Geometry. Graduate Texts in Mathematics, vol. 212. Springer, New York (2002)CrossRefGoogle Scholar
  17. 17.
    Minkowski, H.: Geometrie der Zahlen. Teubner, Leipzig (1910)MATHGoogle Scholar
  18. 18.
    Pach, J., Tóth, G.: Graphs drawn with few crossings per edge. Combinatorica 17(3), 427–439 (1997)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Roth, K.F.: On a problem of Heilbronn. J. Lond. Math. Soc. 26, 198–204 (1951)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Sheffer, A.: Lower bounds for incidences with hypersurfaces. Discrete Anal. 2016, Paper No. 16 (2016)Google Scholar
  21. 21.
    Siegel, C.L., Chandrasekharan, K.: Lectures on the Geometry of Numbers. Springer, Berlin (1989)CrossRefGoogle Scholar
  22. 22.
    Szemerédi, E., Trotter Jr., W.T.: Extremal problems in discrete geometry. Combinatorica 3(3–4), 381–392 (1983)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

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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