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Simple proofs of classical theorems in discrete geometry via the GuthKatz polynomial partitioning technique
 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 10 (5 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.
Distinct distances on two lines
, 2013
"... Let P1 and P2 be two finite sets of points in the plane, so that P1 is contained in a line ℓ1, P2 is contained in a line ℓ2, and ℓ1 and ℓ2 are neither parallel nor orthogonal. Then the number of distinct distances determined by the pairs of P1 ×P2 is Ω ..."
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Let P1 and P2 be two finite sets of points in the plane, so that P1 is contained in a line ℓ1, P2 is contained in a line ℓ2, and ℓ1 and ℓ2 are neither parallel nor orthogonal. Then the number of distinct distances determined by the pairs of P1 ×P2 is Ω
Polynomial partitioning on varieties of codimension two and pointhypersurface incidences in four dimensions, ArXiv eprints
, 2014
"... Abstract. We present a polynomial partitioning theorem for finite sets of points in the real locus of a complex algebraic variety of codimension at most two. This result generalizes the polynomial partitioning theorem on the Euclidean space of Guth and Katz, and its extension to hypersurfaces by Za ..."
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Abstract. We present a polynomial partitioning theorem for finite sets of points in the real locus of a complex algebraic variety of codimension at most two. This result generalizes the polynomial partitioning theorem on the Euclidean space of Guth and Katz, and its extension to hypersurfaces by Zahl and by Kaplan, Matoušek, Sharir and Safernová. We also present a bound for the number of incidences between points and hypersurfaces in the fourdimensional Euclidean space. It is an application of our partitioning theorem together with the refined bounds for the number of connected components of a semialgebraic set by Barone and Basu. Contents
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.
POLYNOMIAL PARTITIONING ON VARIETIES AND POINTHYPERSURFACE INCIDENCES IN FOUR DIMENSIONS
"... Abstract. We present a polynomial partitioning theorem for finite sets of points in the real locus of a complex algebraic variety. This result generalizes the polynomial partitioning theorem on the Euclidean space of Guth and Katz, and its extension to hypersurfaces by Zahl and by Kaplan, Matouˇsek, ..."
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Abstract. We present a polynomial partitioning theorem for finite sets of points in the real locus of a complex algebraic variety. This result generalizes the polynomial partitioning theorem on the Euclidean space of Guth and Katz, and its extension to hypersurfaces by Zahl and by Kaplan, Matouˇsek, Sharir and Safernová. We also present a bound for the number of incidences between points and hypersurfaces in the fourdimensional Euclidean space. It is an application of our partitioning theorem together with the refined bounds for the number of connected components of a semialgebraic set by Barone and Basu. Contents
On lattices, distinct distances, and the ElekesSharir framework
, 2013
"... In this note we consider distinct distances determined by points in an integer lattice. We first consider Erdős’s lower bound for the square lattice, recast in the setup of the socalled ElekesSharir framework [5, 8], and show that, without a major change, this framework cannot lead to Erdős’s conj ..."
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In this note we consider distinct distances determined by points in an integer lattice. We first consider Erdős’s lower bound for the square lattice, recast in the setup of the socalled ElekesSharir framework [5, 8], and show that, without a major change, this framework cannot lead to Erdős’s conjectured lower bound. This shows that the upper bound of Guth and Katz [8] for the related 3dimensional lineintersection problem is tight for this instance. The gap between this bound and the actual bound of Erdős arises from an application of the CauchySchwarz inequality (which is an integral part of the ElekesSharir framework). Our analysis relies on two numbertheoretic results by Ramanujan. We also consider distinct distances in rectangular lattices of the form {(i,j)  0 ≤ i ≤ n 1−α, 0 ≤ j ≤ n α}, for some 0 < α < 1/2, and show that the number of distinct distances in such a lattice is Θ(n). In a sense, our proof “bypasses ” a deep conjecture in number theory, posed by Cilleruelo and Granville [4]. A positive resolution of this conjecture would also have implied our bound.
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
Incidence Theorems and Their Applications
"... We survey recent (and not so recent) results concerning arrangements of lines, points and other geometric objects and the applications these results have in theoretical computer science and combinatorics. The three main types of problems we will discuss are: 1. Counting incidences: Given a set (or s ..."
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We survey recent (and not so recent) results concerning arrangements of lines, points and other geometric objects and the applications these results have in theoretical computer science and combinatorics. The three main types of problems we will discuss are: 1. Counting incidences: Given a set (or several sets) of geometric objects (lines, points, etc.), what is the maximum number of incidences (or intersections) that can exist between elements in different sets? We will see several results of this type, such as the SzemerediTrotter theorem, over the reals and over finite fields and discuss their applications in combinatorics (e.g., in the recent solution of Guth and Katz to Erdos ’ distance problem) and in computer science (in explicit constructions of multisource extractors). 2. Kakeya type problems: These problems deal with arrangements of lines that point in different directions. The goal is to try and understand to what extent these lines can overlap one another. We will discuss these questions both over the reals and over finite fields and see how they come up in the
The Beginnings of Geometric Graph Theory
"... “...to ask the right question and to ask it of the right person.” (Richard Guy) Geometric graphs (topological graphs) are graphs drawn in the plane with possibly crossing straightline edges (resp., curvilinear edges). Starting with a problem of Heinz Hopf and Erika Pannwitz from 1934 and a seminal ..."
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“...to ask the right question and to ask it of the right person.” (Richard Guy) Geometric graphs (topological graphs) are graphs drawn in the plane with possibly crossing straightline edges (resp., curvilinear edges). Starting with a problem of Heinz Hopf and Erika Pannwitz from 1934 and a seminal paper of Paul Erdős from 1946, we give a biased survey of Turántype questions in the theory of geometric and topological graphs. What is the maximum number of edges that a geometric or topological graph of n vertices can have if it contains no forbidden subconfiguration of a certain type? We put special emphasis on open problems raised by Erdős or directly motivated by his work. 1
Improved Bounds for Incidences between Points and Circles
, 2013
"... We establish an improved upper bound for the number of incidences between m points and n arbitrary circles in three dimensions. 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 ∗ (·) ..."
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We establish an improved upper bound for the number of incidences between m points and n arbitrary circles in three dimensions. 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