Incidences between points and circles in three and higher dimensions

We show that the number of incidences between m distinct points and n distinct circles in ℝ³ is O(m^4/7n^17/21 + m^2/3n^2/3 + m + n); the bound is optimal for m ≥ n^3/2. This result extends recent work on point-circle incidences in the plane, but its proof requires a different analysis. The bound improves upon a previous bound, noted by Akutsu et al. [2] and by Agarwal and Sharir [1], but it is not as sharp (when m is small) as the recent planar bound of Aronov and Sharir [3]. Our analysis extends to yield the same bound (a) on the number of incidences between m points and n circles in any dimension d ≥ 3, and (b) on the number of incidences between m points and n arbitrary convex plane curves in ℝ^d, for any d ≥ 3, provided that no two curves are coplanar. Our results improve the upper bound on the number of congruent copies of a fixed tetrahedron in a set of n points in 4-space, and were already used to obtain a lower bound for the number of distinct distances in a set of n points in 3-space.