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38
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 39 (10 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.
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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Cited by 27 (8 self)
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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.
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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Cited by 13 (2 self)
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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.
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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Cited by 7 (3 self)
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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
Polynomials vanishing on grids: The ElekesRónyai problem revisited ∗
, 2014
"... In 2000, Elekes and Rónyai [8] considered the following problem. Let A, B be two sets, each of n real numbers, and let f be a real bivariate polynomial of some constant degree. They showed that if f(A × B)  ≤ cn, for some constant c that depends on deg f, and for n ≥ n0(c), for sufficiently large ..."
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Cited by 6 (2 self)
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In 2000, Elekes and Rónyai [8] considered the following problem. Let A, B be two sets, each of n real numbers, and let f be a real bivariate polynomial of some constant degree. They showed that if f(A × B)  ≤ cn, for some constant c that depends on deg f, and for n ≥ n0(c), for sufficiently large threshold n0(c) that depends on c, then f must be of one of the special forms f(u, v) = h(ϕ(u)+ψ(v)), or f(u, v) = h(ϕ(u)·ψ(v)), for some univariate polynomials ϕ, ψ, h over R. A refined analysis and generalizations of this result are given by Elekes and Szabó [10]. In a variant of this setup, we have, in addition to A, B and f, another set C of n real numbers, and we denote by M the number of triples (a, b, c) ∈ A × B × C such that c = f(a, b). We show that either M = O(n 11/6), where the constant of proportionality depends on deg f, or else f has one of the above special forms. This improves the results in [8, 10], by making the bound explicit, with an exponent that is independent of the degree of f. We actually establish a more general result, for the case where A, B, C  are not
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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Cited by 4 (0 self)
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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.
Incidences between points and lines in three dimensions∗
, 2015
"... We give a fairly elementary and simple proof that shows that the number of incidences between m points and n lines in R3, so that no plane contains more than s lines, is ..."
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Cited by 3 (3 self)
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We give a fairly elementary and simple proof that shows that the number of incidences between m points and n lines in R3, so that no plane contains more than s lines, is