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, which applies in any dimension, is $O^*(m^{2/3}n^{2/3} + m^{6/11}n^{9/11}+m+n)$. Since all the points and circles may lie on a common plane (or sphere), it is impossible to improve the three-dimensional bound without improving the two-dimensional one.
Nevertheless, we show that if the set of circles is required to be "truly three-dimensional" in the sense that there exists a $q < n$ so that no sphere or plane contains more than $q$ of the circles, then the bound can be improved to $O^*(m^{3/7}n^{6/7} + m^{2/3}n^{1/2}q^{1/6} + m^{6/11}n^{15/22}q^{3/22} + m + n)$.
For various ranges of parameters (e.g., when $m = \Theta(n)$ and $q = o(n^{7/9}))$, this bound is smaller than the best known two-dimensional lower bound $\Omega*(m^{2/3}n^{2/3}+m+n)$. Thus we obtain an incidence theorem analogous to the one in the distinct distances paper by Guth and Katz, which states that if we have a collection of points and lines in $\mathbb{R}^3$ and we restrict the number of lines that can lie on a common plane or regulus, then the maximum number of point-line incidences is smaller than the maximum number of incidences that can occur in the plane.
Our result is obtained by applying the polynomial partitioning technique of Guth and Katz using a constant-degree partitioning polynomial, as was also used by Solymosi and Tao. We also rely on various additional tools from analytic, algebraic, and combinatorial geometry.
This is joint work with Micha Sharir and Joshua Zahl.
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