The number of different distances determined by a set of points in the Euclidean plane
Discrete & Computational Geometry
Crossing Numbers and Hard Erdös Problems in Discrete Geometry
Combinatorics, Probability and Computing
Point–Line Incidences in Space
Combinatorics, Probability and Computing
Incidences between Points and Circles in Three and Higher Dimensions
Discrete & Computational Geometry
On a question of bourgain about geometric incidences
Combinatorics, Probability and Computing
Discrete & Computational Geometry
On lines, joints, and incidences in three dimensions
Journal of Combinatorial Theory Series A
Computing the distance between piecewise-linear bivariate functions
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
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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 R3. Applying these bounds, we obtain, among several other results, the upper bound O(s3) 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(S3/k12/7). One of our unresolved conjectures is that this number is O(s3/k2), for k ≥ 2. If true, it would imply the lower bound Ω(s/ log s) on the number of distinct distances in the plane.