## Incidences between points and non-coplanar circles

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.