# On the Number of Incidences Between Points and Planes in Three Dimensions

## Abstract

We prove an incidence theorem for points and planes in the projective space ℙ3 over any Field $$\mathbb{F}$$, whose characteristic p ≠ 2. An incidence is viewed as an intersection along a line of a pair of two-planes from two canonical rulings of the Klein quadric. The Klein quadric can be traversed by a generic hyperplane, yielding a line-line incidence problem in a three-quadric, the Klein image of a regular line complex. This hyperplane can be chosen so that at most two lines meet. Hence, one can apply an algebraic theorem of Guth and Katz, with a constraint involving p if p > 0.

This yields a bound on the number of incidences between m points and n planes in ℙ3, with mn as

$$O\left( {m\sqrt n + mk} \right)$$

, where k is the maximum number of collinear planes, provided that n = O(p2) if p > 0. Examples show that this bound cannot be improved without additional assumptions.

This gives one a vehicle to establish geometric incidence estimates when p >0. For a non-collinear point set S⊆F2 and a non-degenerate symmetric or skew-symmetric bilinear form ω, the number of distinct values of ω on pairs of points of S is $$\Omega \left[ {\min \left( {{{\left| S \right|}^{\frac{2}{3}}},p} \right)} \right]$$. This is also the best known bound over ℝ, where it follows from the Szemerédi-Trotter theorem. Also, a set S ⊆ F3, not supported in a single semi-isotropic plane contains a point, from which $$\Omega \left[ {\min \left( {{{\left| S \right|}^{\frac{1}{2}}},p} \right)} \right]$$ distinct distances to other points of S are attained.

This is a preview of subscription content, access via your institution.

## References

1. 

R. Apfelbaum and M. Sharir: Large complete bipartite subgraphs in incidence graphs of points and hyperplanes, SIAM J. Discrete Math. 21 (2007), 707–725.

2. 

E. A. Yazici, B. Murphy, M. Rudnev and I. D. Shkredov: Growth Estimates in Positive Characteristic via Collisions, Int Math Res Notices, first published online October 28, 2016 doi:10.1093/imrn/rnw206.

3. 

L. Badescu: Algebraic Surfaces, Springer, New York, 2001.

4. 

A. Basit and A. Sheffer: Incidences with k-non-degenerate Sets and Their Applications, J. Computational Geometry. 51 (2014), 284–302.

5. 

B. Bekka and M. Mayer: Ergodic theory and topological dynamics of group actions on homogeneous spaces, London Mathematical Society Lecture Note Series, 269, Cambridge University Press, Cambridge, 2000. 200pp.

6. 

T. Bloom and T. G. F. Jones: A sum-product theorem in function fields, Int. Math. Res. Not. IMRN 19 (2014), 5249–5263.

7. 

J. Bourgain, N. Katz and T. Tao: A sum-product estimate in finite fields, and applications, Geom. Funct. Anal. 14 (2004), 27–57.

8. 

P. Brass and C. Knauer: On counting point-hyperplane incidences, Comput. Geom. 25 (2003), 13–20.

9. 

Z. Dvir: On the size of Kakeya sets in finite fields J. Amer. Math. Soc. 22 (2009), 1093–1097.

10. 

Z. Dvir: Incidence Theorems and Their Applications, Preprint arXiv:1208.5073v2 [math.CO] 27 Aug 2013. Survey 104pp.

11. 

L. Edelsbrunner, L. Guibas and M. Sharir: The complexity of many cells in arrangements of planes and related problems, Discrete Comput. Geom. 5 (1990), 197–216.

12. 

G. Elekes: On the number of sums and products, Acta Arith. 81 (1997), 365–367.

13. 

G. Elekes and C. Tóth: Incidences of not-too-degenerate hyperplanes, Computational geometry (SCG’05), 16–21, ACM, New York, 2005.

14. 

J. S. Ellenberg and M. Hablicsek: An incidence conjecture of Bourgain over fields of positive characteristic, Preprint arXiv:1311.1479 [math.CO] 6 Nov 2013.

15. 

L. Guth and N. H. Katz: On the Erdos distinct distance problem in the plane. Ann. of Math. (2) 181 (2015), 155–190.

16. 

L. Guth and N. H. Katz: Algebraic methods in discrete analogs of the Kakeya problem, Adv. Math. 225 (2010), 2828–2839.

17. 

D. Hart, A. Iosevich, D. Koh and M. Rudnev: Averages over hyperplanes, sumproduct theory in vector spaces over finite fields and the Erdos-Falconer distance conjecture, Trans. Amer. Math. Soc. 363 (2011), 3255–3275.

18. 

D. R. Heath-Brown and S. V. Konyagin: New bounds for Gauss sums derived from kth powers, and for Heilbronn’s exponential sum, Q. J. Math. 51 (2000), 221–235.

19. 

A. Iosevich, S. Konyagin, M. Rudnev and V. Ten: Combinatorial complexity of convex sequences, Discrete Comput. Geom. 35 (2006), 143–158.

20. 

A. Iosevich, O. Roche-Newton, and M. Rudnev: On an application of the GuthKatz Theorem, Math. Res. Lett. 18 (2011), 691–697.

21. 

A. Iosevich, O. Roche-Newton and M. Rudnev: On discrete values of bilinear forms, Preprint arXiv:1512.0267 [math.CO] 8 Dec 2015.

22. 

T. G. F. Jones: Further improvements to incidence and Beck-type bounds over prime fields, Preprint arXiv:1206.4517 [math.CO] 20 Jun 2012.

23. 

C. Liedtke: Algebraic Surfaces in Positive Characteristic, in: Birational Geometry, Rational Curves, and Arithmetic, Springer, 2013, 229–292.

24. 

N. H. Katz: The flecnode polynomial: a central object in incidence geometry, Preprint arXiv:1404.3412 [math.CO] 13 Apr 2014.

25. 

J. Kollár: Szemerédi-Trotter-type theorems in dimension 3, Adv. Math. 271 (2015), 30–61.

26. 

S. V. Konyagin and M. Rudnev: On new sum-product type estimates, SIAM J. Discrete Math. 27 (2013), 973–990.

27. 

S. V. Konyagin and I. D. Shkredov: On sum sets of sets, having small product set, Tr. Mat. Inst. Steklova 290 (2015), 304–316.

28. 

J. Plücker: Neue Geometrie des Raumes, gegrundet auf die Betrachtung der geraden Linie als Raumelement, 2 vols., Leipzig: B. G. Teubner, 1868–1869.

29. 

H. Pottmann and J. Wallner: Computational Line Geometry, Springer Verlag, Berlin, 2001.

30. 

O. Roche-Newton and M. Rudnev: On the Minkowski distances and products of sum sets, Israel J. Math. 209 (2015), 507–526.

31. 

O. Roche-Newton, M. Rudnev and I. D. Shkredov: New sum-product type estimates over finite fields, Adv. Math. 293 (2016), 589–605.

32. 

M. Rudnev: An Improved Sum-Product Inequality in Fields of Prime Order, Int. Math. Res. Not. IMRN 16 (2012), 3693–3705.

33. 

M. Rudnev and J. M. Selig: On the use of Klein quadric for geometric incidence problems in two dimensions, SIAM Journal on Discrete Mathematics 30 (2016), 934–954.

34. 

G. Salmon: A treatise on the analytic geometry of three dimensions, vol. 2, 5th edition, Longmans, Green and Co., London 1915.

35. 

J. M. Selig: Geometric Fundamentals of Robotics, Monographs in Computer Science, Springer, 2007.

36. 

J. Solymosi: Bounding multiplicative energy by the sumset, Adv. Math. 222 (2009), 402–408.

37. 

J. Solymosi and V. H. Vu: Near optimal bounds for the Erdos distinct distances problem in high dimensions, Combinatorica 28 (2008), 113–125.

38. 

E. Szemerédi and W. T. Trotter: Extremal problems in discrete geometry, Combinatorica 3 (1983), 381–392.

39. 

C. Tóth: The Szemerédi-Trotter theorem in the complex plane, Combinatorica 3 (2015), 95–126.

40. 

F. Voloch: Surfaces in P3 over finite fields, Topics in algebraic and noncommutative geometry (Luminy/Annapolis, MD, 2001), 219–226, Contemp. Math. 324, Amer. Math. Soc., Providence, RI, 2003.

41. 

I. V. V’yugin and I. D. Shkredov: On additive shifts of multiplicative subgroups, Mat. Sb. 203 (2012), 81–100 (in Russian).

Download references

## Author information

Authors

### Corresponding author

Correspondence to Misha Rudnev.

## Rights and permissions

Reprints and Permissions