Incidences in Three Dimensions and Distinct Distances in the Plane

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 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. Categories and Subject Descriptors 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...

11. Point-plane incidences and some applications in positive characteristic

“…, which, although vacuous for |S| ≤ p, beats the claim of Theorem 12 already for |S| ≥ p 30 29 . The key quantity behind the latter estimate is the energy-type non-pinned version of equation (14), that is the variable s in the right-hand side of the three-variable equation in (14) turns into the fourth variable s ′ ∈ S. For the minimum number of pinned distances for |S| ≥ p 15 14 in F p one can use estimate (6), which yields the existence of Ω Sketch of proof of Theorem 12. Consider equation (16), assuming that at most ǫ|S| points of S are collinear, for some absolute ǫ > 0, or there is nothing to prove.…”

An L2-identity and pinned distance problem

“…The proof relies on a group action argument. The idea of using group action argument to attack distance problem dates back to the solution to the Erdős distance problem ( [8], [12]). On Falconer distance problem, authors in [11] observed that Mattila's integral (1.1) can be interpreted in terms of Haar measures on O(d), that is,…”

A Refined Energy Bound for Distinct Perpendicular Bisectors

“…First, if (S ac , (b, d)) ∈ G, then |ac| = |bd|. This is easy to derive from equations (3) and (4): expand the products and subtract (4) from (3). Also see Figure 4.…”

“…The same paper includes the conjecture that the strongest possible bound is 1 The term additive energy, referring to the number of quadruples (a, b, c, d) in some underlying set of numbers such that a + b = c + d, was coined by Tao and Vu [8]. Starting with the work of Elekes and Sharir [3], and Guth and Katz [6] on the distinct distance problem, the strategy of using geometric incidence bounds to obtain upper bounds on analogously defined energies has become indispensable in the study of questions about the number of distinct equivalent subsets. |Q| = O(Kn 2 ).…”