We show that the number of incidences between m distinct points and n distinct circles in R d, for any d ≥ 3, is O(m 6/11 n 9/11 κ(m 3 /n)+m 2/3 n 2/3 +m+n), where κ(n) = (log n) O(α2 (n)) 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(m 4/7 n 17/21 + m 2/3 n 2/3 + 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.

Let L be a set of n lines in space. A joint of L is a point in R 3 where at least three non-coplanar lines meet. We show that the number of joints of L is O(n

Abstract. We prove the joints conjecture, showing that for any N lines in R 3, there are at most O(N 3 2) points at which 3 lines intersect non-coplanarly. We also prove a conjecture of Bourgain showing that given N 2 lines in R 3 so that no N lines lie in the same plane and so that each line intersects a set P of points in at least N points then the cardinality of the set of points is Ω(N 3). Both our proofs are adaptations of Dvir’s argument for the finite field Kakeya problem. 1.

We first describe a reduction from the problem of lower-bounding the number of distinct distances determined by a set S of s points in the plane to an incidence problem between points and a certain class of helices (or parabolas) in three dimensions. We offer conjectures involving the new setup, but are still unable to fully resolve them. Instead, we adapt the recent new algebraic analysis technique of Guth and Katz [9], as further developed by Elekes et al. [6], to obtain sharp bounds on the number of incidences between these helices or parabolas and points in R 3. Applying these bounds, we obtain, among several other results, the upper bound O(s 3) on the number of rotations (rigid motions) which map (at least) three points of S to three other points of S. In fact, we show that the number of such rotations which map at least k ≥ 3 points of S to k other points of S is close to O(s 3 /k 12/7). One of our unresolved conjectures is that this number is O(s 3 /k 2), for k ≥ 2. If true, it would imply the lower bound Ω(s / log s) on the number of distinct distances in the plane.