Improved bounds for incidences between points and circles

• Authors: Micha Sharir;Adam Sheffer;Joshua Zahl
• Venue: Proceedings of the twenty-ninth annual symposium on Computational geometry
• Year: 2013

We establish an improved upper bound for the number of incidences between m points and n arbitrary 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 sub-polynomial factors). Since all the points and circles may lie on a common plane or sphere, it is impossible to improve the bound in R^3 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 q 3/7n6/7 + m2/3n1/2q1/6 + m6/11n15/22q3/22 + m + n). For various ranges of parameters (e.g., when m=Θ(n) and q = o(n7/9)), this bound is smaller than the best known two-dimensional worst-case lower bound Ω*(m2/3n2/3+m+n). We present several extensions and applications of the new bound: (i) For the special case where all the circles have the same radius, we obtain the improved bound O*(m5/11n9/11 + m2/3n1/2q1/6 + m + n). (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. (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.