TY - JOUR

T1 - Incidences in three dimensions and distinct distances in the plane

AU - Elekes, György

AU - Sharir, Micha

PY - 2011/7

Y1 - 2011/7

N2 - 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 set-up, 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, Kaplan and Sharir [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/logs) on the number of distinct distances in the plane.

AB - 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 set-up, 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, Kaplan and Sharir [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/logs) on the number of distinct distances in the plane.

UR - http://www.scopus.com/inward/record.url?scp=79958856267&partnerID=8YFLogxK

U2 - 10.1017/S0963548311000137

DO - 10.1017/S0963548311000137

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AN - SCOPUS:79958856267

SN - 0963-5483

VL - 20

SP - 571

EP - 608

JO - Combinatorics Probability and Computing

JF - Combinatorics Probability and Computing

IS - 4

ER -