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Incidences in Three Dimensions and Distinct Distances in the Plane

Published online by Cambridge University Press:  04 April 2011

GYÖRGY ELEKES
Affiliation:
Department of Computer Science, Eötvös University, Budapest, Hungary
MICHA SHARIR
Affiliation:
School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel and Courant Institute of Mathematical Sciences, New York University, NY 10012, USA (e-mail: michas@post.tau.ac.il)
Corresponding
E-mail address:

Abstract

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.

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Copyright
Copyright © Cambridge University Press 2011

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