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Improved Bounds for Incidences Between Points and Circles

Published online by Cambridge University Press:  02 October 2014

MICHA SHARIR
Affiliation:
School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: michas@tau.ac.il, sheffera@tau.ac.il)
ADAM SHEFFER
Affiliation:
School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel (e-mail: michas@tau.ac.il, sheffera@tau.ac.il)
JOSHUA ZAHL
Affiliation:
Department of Mathematics, UCLA, Los Angeles, CA 90024, USA (e-mail: jzahl@mit.edu)
Corresponding

Abstract

We establish an improved upper bound for the number of incidences between m points and n circles in three dimensions. The previous best known bound, originally established for the planar case and later extended to any dimension ≥ 2, is O*(m2/3n2/3 + m6/11n9/11 + m + n), where the O*(⋅) notation hides polylogarithmic factors. Since all the points and circles may lie on a common plane (or sphere), it is impossible to improve the bound in ℝ3 without first improving it in the plane.

Nevertheless, we show that if the set of circles is required to be ‘truly three-dimensional’ in the sense that no sphere or plane contains more than q of the circles, for some qn, then for any ϵ > 0 the bound can be improved to

\[ O\bigl(m^{3/7+\eps}n^{6/7} + m^{2/3+\eps}n^{1/2}q^{1/6} + m^{6/11+\eps}n^{15/22}q^{3/22} + m + n\bigr). \]
For various ranges of parameters (e.g., when m = Θ(n) and q = o(n7/9)), this bound is smaller than the lower bound Ω*(m2/3n2/3 + m + n), which holds in two dimensions.

We present several extensions and applications of the new bound.

  1. (i) For the special case where all the circles have the same radius, we obtain the improved bound O(m5/11+ϵn9/11 + m2/3+ϵn1/2q1/6 + m + n).

  2. (ii) We present an improved analysis that removes the subpolynomial factors from the bound when m = O(n3/2−ϵ) for any fixed ϵ < 0.

  3. (iii) We use our results to obtain the improved bound O(m15/7) for the number of mutually similar triangles determined by any set of m points in ℝ3.

Our result is obtained by applying the polynomial partitioning technique of Guth and Katz using a constant-degree partitioning polynomial (as was also recently used by Solymosi and Tao). We also rely on various additional tools from analytic, algebraic, and combinatorial geometry.

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Copyright
Copyright © Cambridge University Press 2014 

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References

[1]Agarwal, P. K., Apfelbaum, R., Purdy, G. and Sharir, M. (2007) Similar simplices in a d-dimensional point set. In Proc. 23rd ACM Symposium on Computational Geometry, pp. 232–238.Google Scholar
[2]Agarwal, P. K., Matoušek, J. and Sharir, M. (2013) On range searching with semialgebraic sets~II. SIAM J. Comput. 42 20392062.CrossRefGoogle Scholar
[3]Agarwal, P., Nevo, E., Pach, J., Pinchasi, R., Sharir, M. and Smorodinsky, S. (2004) Lenses in arrangements of pseudocircles and their applications. J. Assoc. Comput. Mach. 51 139186.CrossRefGoogle Scholar
[4]Akutsu, T., Tamaki, H. and Tokuyama, T. (1998) Distribution of distances and triangles in a point set and algorithms for computing the largest common point sets. Discrete Comput. Geom. 20 307331.CrossRefGoogle Scholar
[5]Alon, N. (1995) Tools from higher algebra. In Handbook of Combinatorics, Vols~1, 2, Elsevier, pp. 17491783.Google Scholar
[6]Apfelbaum, R. and Sharir, M. (2011) Non-degenerate spheres in three dimensions. Combin. Probab. Comput. 20 503512.CrossRefGoogle Scholar
[7]Aronov, B., Koltun, V. and Sharir, M. (2005) Incidences between points and circles in three and higher dimensions, Discrete Comput. Geom. 33 185206.CrossRefGoogle Scholar
[8]Aronov, B. and Sharir, M. (2002) Cutting circles into pseudo-segments and improved bounds for incidences. Discrete Comput. Geom. 28 475490.CrossRefGoogle Scholar
[9]Basu, S. and Sombra, M. Polynomial partitioning on varieties and point–hypersurface incidences in four dimensions. arXiv:1406.2144.Google Scholar
[10]Beauville, A. (1996) Complex Algebraic Surfaces, second edition, Vol. 34 of London Mathematical Society Student Texts, Cambridge University Press.CrossRefGoogle Scholar
[11]Bochnak, J., Coste, M. and Roy, M. (1998) Real Algebraic Geometry, Springer.CrossRefGoogle Scholar
[12]Brass, P. (2002) Combinatorial geometry problems in pattern recognition. Discrete Comput. Geom. 28 495510.CrossRefGoogle Scholar
[13]Chazelle, B., Edelsbrunner, H., Guibas, L. J., Sharir, M. and Stolfi, J. (1996) Lines in space: Combinatorics and algorithms. Algorithmica 15 428447.CrossRefGoogle Scholar
[14]Cox, D., Little, J. and O'Shea, D. (2004) Using Algebraic Geometry, second edition, Springer.Google Scholar
[15]Cox, D., Little, J. and O'Shea, D. (2007) Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, third edition, Springer.CrossRefGoogle Scholar
[16]Elekes, G., Kaplan, H. and Sharir, M. (2011) On lines, joints, and incidences in three dimensions. J. Combin. Theory Ser. A 118 962977.CrossRefGoogle Scholar
[17]Elekes, G. and Sharir, M. (2011) Incidences in three dimensions and distinct distances in the plane. Combin. Probab. Comput. 20 571608.CrossRefGoogle Scholar
[18]Erdős, P. (1946) On sets of distances of n points. Amer. Math. Monthly 53 248250.CrossRefGoogle Scholar
[19]Fox, J., Pach, J., Sheffer, A., Suk, A. and Zahl, J. A semi-algebraic version of Zarankiewicz's problem, arXiv:1407.5705.Google Scholar
[20]Fuchs, D. and Tabachnikov, S. (2007) Mathematical Omnibus: Thirty Lectures on Classical Mathematics, AMS.CrossRefGoogle Scholar
[21]Fulton, W. (1998) Intersection Theory, Springer.CrossRefGoogle Scholar
[22]Guth, L. Distinct distance estimates and low degree polynomial partitioning. arXiv:1404.2321.Google Scholar
[23]Guth, L. and Katz, N. H. (2010) Algebraic methods in discrete analogs of the Kakeya problem. Adv. Math. 225 28282839.CrossRefGoogle Scholar
[24]Guth, L. and Katz, N. H. On the Erdős distinct distances problem in the plane. Ann. of Math., to appear. arXiv:1011.4105.Google Scholar
[25]Harris, J. (1992) Algebraic Geometry: A First Course, Springer.CrossRefGoogle Scholar
[26]Hartshorne, R. (1983) Algebraic Geometry, Springer.Google Scholar
[27]Hartshorne, R. (2000) Geometry: Euclid and Beyond, Springer.CrossRefGoogle Scholar
[28]Hwang, J. M. (2005) A bound on the number of curves of a given degree through a general point of a projective variety. Compositio Math. 141 703712.CrossRefGoogle Scholar
[29]Kaplan, H., Matoušek, J., Safernová, Z. and Sharir, M. (2012) Unit distances in three dimensions. Combin. Probab. Comput. 21 597610.CrossRefGoogle Scholar
[30]Kaplan, H., Matoušek, J. and Sharir, M. (2012) Simple proofs of classical theorems in discrete geometry via the Guth–Katz polynomial partitioning technique. Discrete Comput. Geom. 48 499517.CrossRefGoogle Scholar
[31]Kaplan, H., Sharir, M. and Shustin, E. (2010) On lines and joints. Discrete Comput. Geom. 44 838843.CrossRefGoogle Scholar
[32]Landsberg, J. M. (1999) Is a linear space contained in a submanifold? On the number of derivatives needed to tell. J. Reine Angew. Math. 508 5360.CrossRefGoogle Scholar
[33]Landsberg, J. M. (2003) Lines on projective varieties. J. Reine Angew. Math. 562 13.CrossRefGoogle Scholar
[34]Marcus, A. and Tardos, G. (2006) Intersection reverse sequences and geometric applications. J. Combin. Theory Ser. A 113 675691.CrossRefGoogle Scholar
[35]Matoušek, J. (2002) Lectures on Discrete Geometry, Springer.CrossRefGoogle Scholar
[36]Milnor, J. (1964) On the Betti numbers of real varieties. Proc. Amer. Math. Soc. 15 275280.CrossRefGoogle Scholar
[37]Miranda, R. (1995) Algebraic Curves and Riemann Surfaces, Vol. 5 of Graduate Studies in Mathematics, AMS.CrossRefGoogle Scholar
[38]Nilov, F. and Skopenkov, M. (2013) A surface containing a line and a circle through each point is a quadric. Geom. Dedicata 163 301310.CrossRefGoogle Scholar
[39]Pach, J. and Sharir, M. (2004) Geometric incidences. In Towards a Theory of Geometric Graphs (Pach, J., ed.), Vol. 342 of Contemporary Mathematics, AMS, pp. 185223.CrossRefGoogle Scholar
[40]Quilodrán, R. (2010) The joints problem in Rn. SIAM J. Discrete Math. 23 22112213.CrossRefGoogle Scholar
[41]Roy, M.-F. and Vorobjov, N. (2002) The complexification and degree of a semi-algebraic set. Math. Z. 239 131142.CrossRefGoogle Scholar
[42]Salmon, G. (1915) A Treatise on the Analytic Geometry of Three Dimensions, Vol. 2, fifth edition, Hodges, Figgis and Co. Ltd., Dublin.Google Scholar
[43]Sard, A. (1942) The measure of the critical values of differentiable maps. Bull. Amer. Math. Soc. 48 883890.CrossRefGoogle Scholar
[44]Sharir, M. and Solomon, N. (2014) Incidences between points and lines in four dimensions. In Proc. 30th ACM Symposium on Computational Geometry, 189–197.Google Scholar
[45]Solymosi, J. and Tao, T. (2012) An incidence theorem in higher dimensions. Discrete Comput. Geom. 48 255280.CrossRefGoogle Scholar
[46]Székely, L. (1997) Crossing numbers and hard Erdős problems in discrete geometry. Combin. Probab. Comput. 6 353358.CrossRefGoogle Scholar
[47]Thom, R. (1965) Sur l'homologie des variétés algebriques réelles. In Differential and Combinatorial Topology (Cairns, S. S., ed.), Princeton University Press, pp. 255265.Google Scholar
[48]Warren, H. E. (1968) Lower bound for approximation by nonlinear manifolds. Trans. Amer. Math. Soc. 133 167178.CrossRefGoogle Scholar
[49]Whitney, H. (1957) Elementary structure of real algebraic varieties. Ann. of Math. 66 545556.CrossRefGoogle Scholar
[50]Zahl, J. (2013) An improved bound on the number of point–surface incidences in three dimensions. Contrib. Discrete Math. 8 100121.Google Scholar
[51]Zahl, J. A Szemerédi–Trotter type theorem in ℝ4. arXiv:1203.4600.Google Scholar
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