Incidences between points and circles in three and higher dimensions

(MATH) We show that the number of incidences between <i>m</i> distinct points and <i>n</i> distinct circles in $\reals^3$ is <i>O</i>(<i>m</i> <sup>4/7</sup> <i>n</i> <sup>17/21</sup>+<i>m</i> <sup>2/3</sup> <i>n</i> <sup>2/3</sup>+<i>m</i>+<i>n</i>); the bound is optimal for <i>m n</i> <sup>3/2</sup>. 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 <i>m</i> 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 <i>m</i> points and <i>n</i> circles in any dimension <i>d</i>&rhoe; 3, and (b) on the number of incidences between <i>m</i> points and <i>n</i> arbitrary convex plane curves in $\reals^d$, for any <i>d</i>&rhoe; 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 <i>n</i> points in 4-space, and were already used to obtain a lower bound for the number of distinct distances in a set of <i>n</i> points in 3-space.

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