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13
Incidences in Three Dimensions and Distinct Distances in the Plane (Extended Abstract)
, 2010
"... We first describe a reduction from the problem of lowerbounding 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 ..."
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We first describe a reduction from the problem of lowerbounding 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.
Simple proofs of classical theorems in discretegeometryviatheGuthKatzpolynomialpartitioningtechnique
 Discrete Comput. Geom
"... Recently Guth and Katz [16] invented, as a step in their nearly complete solution of Erdős’s distinct distances problem, a new method for partitioning finite point sets in R d, based on the Stone–Tukey polynomial hamsandwich theorem. We apply this method to obtain new and simple proofs of two well ..."
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Cited by 8 (4 self)
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Recently Guth and Katz [16] invented, as a step in their nearly complete solution of Erdős’s distinct distances problem, a new method for partitioning finite point sets in R d, based on the Stone–Tukey polynomial hamsandwich theorem. We apply this method to obtain new and simple proofs of two well known results: the Szemerédi–Trotter theorem on incidences of points and lines, and the existence of spanning trees with low crossing numbers. Since we consider these proofs particularly suitable for teaching, we aim at selfcontained, expository treatment. We also mention some generalizations and extensions, such as the Pach–Sharir bound on the number of incidences with algebraic curves of bounded degree. 1
On Lines and Joints
, 2009
"... Let L be a set of n lines in R d, for d ≥ 3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum possible number of joints of L is Θ(n d/(d−1)). For d = 3, this is a considerable simplifica ..."
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Let L be a set of n lines in R d, for d ≥ 3. A joint of L is a point incident to at least d lines of L, not all in a common hyperplane. Using a very simple algebraic proof technique, we show that the maximum possible number of joints of L is Θ(n d/(d−1)). For d = 3, this is a considerable simplification of the orignal algebraic proof of Guth and Katz [9], and of the followup simpler proof of Elekes et al. [6]. Some extensions, e.g., to the case of joints of algebraic curves, are also presented.
On range searching with semialgebraic sets ii
 In Foundations of Computer Science (FOCS), 2012 IEEE 53rd Annual Symposium on
, 2012
"... Let P be a set of n points in Rd. We present a linearsize data structure for answering range queries on P with constantcomplexity semialgebraic sets as ranges, in time close to O(n1−1/d). It essentially matches the performance of similar structures for simplex range searching, and, for d ≥ 5, sign ..."
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Let P be a set of n points in Rd. We present a linearsize data structure for answering range queries on P with constantcomplexity semialgebraic sets as ranges, in time close to O(n1−1/d). It essentially matches the performance of similar structures for simplex range searching, and, for d ≥ 5, significantly improves earlier solutions by the first two authors obtained in 1994. This almost settles a longstanding open problem in range searching. The data structure is based on the polynomialpartitioning technique of Guth and Katz [arXiv:1011.4105], which shows that for a parameter r, 1 < r ≤ n, there exists a dvariate polynomial f of degree O(r1/d) such that each connected component of Rd \ Z(f) contains at most n/r points of P, where Z(f) is the zero set of f. We present an efficient randomized algorithm for computing such a polynomial partition, which is of independent interest and is likely to have additional applications.
Computing the Distance between PiecewiseLinear Bivariate Functions
"... We consider the problem of computing the distance between two piecewiselinear bivariate functions f and g defined over a common domain M. We focus on the distance induced by the L2norm, that is ‖f − g‖2 = M (f − g)2. If f is defined by linear interpolation over a triangulation of M with n triangle ..."
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We consider the problem of computing the distance between two piecewiselinear bivariate functions f and g defined over a common domain M. We focus on the distance induced by the L2norm, that is ‖f − g‖2 = M (f − g)2. If f is defined by linear interpolation over a triangulation of M with n triangles, while g is defined over another such triangulation, the obvious naïve algorithm requires Θ(n 2) arithmetic operations to compute this distance. We show that it is possible to compute it in O(n log 4 n) arithmetic operations, by reducing the problem to multipoint evaluation of a certain type of polynomials. We also present an application to terrain matching.
The Dawn of an Algebraic . . .
, 2011
"... To me, 2010 looks as annus mirabilis, a miraculous year, in several areas of my mathematical interests. Below I list seven highlights and breakthroughs, mostly in discrete geometry, hoping to share some of my wonder and pleasure with the readers. Of course, hardly any of these great results have com ..."
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To me, 2010 looks as annus mirabilis, a miraculous year, in several areas of my mathematical interests. Below I list seven highlights and breakthroughs, mostly in discrete geometry, hoping to share some of my wonder and pleasure with the readers. Of course, hardly any of these great results have come out of the blue: usually the paper I refer to adds the last step to earlier ideas. Since this is an extended abstract (of a nonexistent paper), I will be rather brief, or sometimes completely silent, about the history, with apologies to the unmentioned giants on whose shoulders the authors I do mention have been standing. 1 A careful reader may notice that together with these great results, I will also advertise some smaller results of mine. • Larry Guth and Nets Hawk Katz [16] completed a bold project of György Elekes (whose previous stage is reported in [10]) and obtained a neartight bound for the Erdős distinct distances problem: they proved that every n points in the plane determine at least Ω(n / log n) distinct distances. This almost matches the best known upper bound of O(n / √ √ √ log n), attained for the n × n grid. Their proof and some related results and methods constitute the main topic of this note, and will be discussed later. • János Pach and Gábor Tardos [27] found tight lower bounds for the size of εnets for geometric set systems. 2 It has been known for a long time
THE JOINTS PROBLEM IN n
, 906
"... n Abstract. We show that given a collection of A lines in, n � 2, the maximum number of their joints (points incident to at least n lines whose directions form a linearly independent set) is O(An/(n−1)). An analogous result for smooth algebraic curves is also proven. 1. ..."
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n Abstract. We show that given a collection of A lines in, n � 2, the maximum number of their joints (points incident to at least n lines whose directions form a linearly independent set) is O(An/(n−1)). An analogous result for smooth algebraic curves is also proven. 1.
THE JOINTS PROBLEM IN n
, 906
"... Abstract. We show that given a collection of A lines in n, n � 2, the maximum number of incidences between the set of lines and k of their joints (points incident to at least n lines whose directions form a linearly independent set) is O(k 1/n A), thus generalizing to higher dimensions a result in [ ..."
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Abstract. We show that given a collection of A lines in n, n � 2, the maximum number of incidences between the set of lines and k of their joints (points incident to at least n lines whose directions form a linearly independent set) is O(k 1/n A), thus generalizing to higher dimensions a result in [2]. In particular the maximum number of incidences and the maximum number of joints determined by the set of lines is O(A n/(n−1)). Analogous results for smooth algebraic curves are also proven. 1.
Improved Bounds for Incidences between Points and Circles ∗
"... We establish an improved upper bound for the number of incidencesbetweenmpointsandnarbitrarycirclesinthreedimensions. The previous best known bound, originally established for the planar case and later extended to any dimension ≥ 2, is O ∗ m 2/3 n 2/3 +m 6/11 n 9/11 +m+n (where the O ∗ (·) notation ..."
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We establish an improved upper bound for the number of incidencesbetweenmpointsandnarbitrarycirclesinthreedimensions. The previous best known bound, originally established for the planar case and later extended to any dimension ≥ 2, is O ∗ m 2/3 n 2/3 +m 6/11 n 9/11 +m+n (where the O ∗ (·) notation hides subpolynomial factors). Since all the points and circles may lie on a common plane or sphere, it is impossible to improve the bound in R 3 without first improving it in the plane. Nevertheless, we show that if the set of circles is required to be “truly threedimensional”in the sense that no sphere or plane contains more than q of the circles, for some q ≪ n, then the bound can be improved to O ∗ ( m 3/7 n 6/7 +m 2/3 n 1/2 q 1/6 +m 6/11 n 15/22 q 3/22 +m+n). For various ranges of parameters (e.g., when m = Θ(n) and q = o(n 7/9)), this bound is smaller than the best known twodimensional worstcase lower bound Ω ∗ (m 2/3 n 2/3 +m+n). We present several extensions and applications of the new bound: (i) For the special case where all the circles have the same radius, we obtain the improved bound