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 ℝd, for any d ≥ 3, is O(m6/11n9/11κ(m3/n)+m2/3n2/3+m+n), where κ(n)=(log n)O(α(n))2 and where α(n) is the inverse Ackermann function. The bound coincides with the recent bound of Aronov and Sharir, or rather with its slight improvement by Agarwal et al., for the planar case. We also show that the number of incidences between m points and n unrestricted convex (or bounded-degree algebraic) plane curves, no two in a common plane, is O(m4/7n17/21+m2/3n2/3+m+n), in any dimension d ≥ 3. 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 the lower bound for the number of distinct distances in a set of n points in 3-space. Another application is an improved bound for the number of incidences (or, rather, containments) between lines and reguli in three dimensions. The latter result has already been applied by Feldman and Sharir to obtain a new bound on the number of joints in an arrangement of lines in three dimensions.