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. Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. SCG'10, June 13-16, 2010, Snowbird, Utah, USA. Copyright 2010 ACM 978-1-4503-0016-2/10/06 ...$10.00.
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THE INFRASTRUCTUREThe motivation for the study reported in this paper comes from the celebrated and long-standing problem, originally posed by Erdős [8] in 1946, of obtaining a sharp lower bound for the number of distinct distances guaranteed to exist in any set S of s points in the plane. Erdős has shown that a section of the integer lattice determines only O(s/ √ log s) distinct distances, and conjectured this to be a lower bound for any planar point set. Katz and Tardos [11], who obtained the current record of Ω(s (48−14e)/(55−16e)−ε ), for any ε > 0, which is Ω(s 0.8641 ). In this paper we transform the problem of distinct distances in the plane to an incidence problem between points and a certain kind of curves (helices or parabolas) in three dimensions. As we show, sharp upper bounds on the number of such incidences translate back to sharp lower bounds on the number of distinct distances. Incidence problems in three dimensions between points and curves have been studied in several recent works [2,6,16], and a major push in this direction has been made last year, with the breakthrough result of Guth and Katz [9], who have introduced methods from algebraic geometry for studying problems of this kind. This has been picked up by the authors [6], where worst-case tight bounds on the number of incidences between points and lines in three dimensions (under certain restrictions) have been obtaine...
