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Algebraic methods in discrete analogs of the Kakeya problem
, 2008
"... Abstract. We prove the joints conjecture, showing that for any N lines in R 3, there are at most O(N 3 2) points at which 3 lines intersect noncoplanarly. We also prove a conjecture of Bourgain showing that given N 2 lines in R 3 so that no N lines lie in the same plane and so that each line inters ..."
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Cited by 22 (1 self)
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Abstract. We prove the joints conjecture, showing that for any N lines in R 3, there are at most O(N 3 2) points at which 3 lines intersect noncoplanarly. We also prove a conjecture of Bourgain showing that given N 2 lines in R 3 so that no N lines lie in the same plane and so that each line intersects a set P of points in at least N points then the cardinality of the set of points is Ω(N 3). Both our proofs are adaptations of Dvir’s argument for the finite field Kakeya problem. 1.
On the multilinear restriction and Kakeya conjectures
 Acta Math
"... Abstract. We prove dlinear analogues of the classical restriction and Kakeya conjectures in Rd. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We conclude by giving some applications to the correspondin ..."
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Cited by 20 (6 self)
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Abstract. We prove dlinear analogues of the classical restriction and Kakeya conjectures in Rd. Our approach involves obtaining monotonicity formulae pertaining to a certain evolution of families of gaussians, closely related to heat flow. We conclude by giving some applications to the corresponding variablecoefficient problems and the socalled “joints ” problem, as well as presenting some nlinear analogues for n < d. 1.
Incidences between points and circles in three and higher dimensions
 Geom
, 2002
"... We show that the number of incidences between m distinct points and n distinct circles in R d, for any d ≥ 3, is O(m 6/11 n 9/11 κ(m 3 /n)+m 2/3 n 2/3 +m+n), where κ(n) = (log n) O(α2 (n)) and where α(n) is the inverse Ackermann function. The bound coincides with the recent bound of Aronov and Shar ..."
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Cited by 14 (9 self)
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We show that the number of incidences between m distinct points and n distinct circles in R d, for any d ≥ 3, is O(m 6/11 n 9/11 κ(m 3 /n)+m 2/3 n 2/3 +m+n), where κ(n) = (log n) O(α2 (n)) and where α(n) is the inverse Ackermann function. The bound coincides with the recent bound of Aronov and Sharir, or rather with its slight improvement by Agarwal et al., for the planar case. We also show that the number of incidences between m points and n unrestricted convex (or boundeddegree algebraic) plane curves, no two in a common plane, is O(m 4/7 n 17/21 + m 2/3 n 2/3 + m + n), in any dimension d ≥ 3. Our results improve the upper bound on the number of congruent copies of a fixed tetrahedron in a set of n points in 4space and the lower bound for the number of distinct distances in a set of n points in 3space. Another application is an improved bound for the number of incidences (or, rather, containments) between lines and reguli in three dimensions. The latter result has already been applied by Feldman and Sharir to obtain a new bound on the number of joints in an arrangement of lines in three dimensions.
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 6 (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.
On a question of Bourgain about geometric incidences
 Combinat. Probab. Comput
"... Given a set of s points and a set of n 2 lines in threedimensional Euclidean space such that each line is incident to n points but no n lines are coplanar, then we have s = Ω(n 11/4). This is the first nontrivial answer to a question recently posed by Jean Bourgain. 1 ..."
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Cited by 4 (0 self)
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Given a set of s points and a set of n 2 lines in threedimensional Euclidean space such that each line is incident to n points but no n lines are coplanar, then we have s = Ω(n 11/4). This is the first nontrivial answer to a question recently posed by Jean Bourgain. 1
From randomness extraction to rotating needles
 SIGACT News
"... The finite field Kakeya problem deals with the way lines in different directions can overlap in a vector space over a finite field. This problem came up in the study of certain Euclidean problems and, independently, in the search for explicit randomness extractors. We survey recent progress on this ..."
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The finite field Kakeya problem deals with the way lines in different directions can overlap in a vector space over a finite field. This problem came up in the study of certain Euclidean problems and, independently, in the search for explicit randomness extractors. We survey recent progress on this problem and describe several of its applications. 1
On Distinct Distances and Incidences: Elekes’s Transformation and the New Algebraic Developments
, 2010
"... We first present a transformation that Gyuri Elekes has devised, about a decade ago, from the celebrated problem of Erdős 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 parabo ..."
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We first present a transformation that Gyuri Elekes has devised, about a decade ago, from the celebrated problem of Erdős 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. Elekes has offered conjectures involving the new setup, which, if correct, would imply that the number of distinct distances in an selement point set in the plane is always Ω(s/log s). Unfortunately, these conjectures are still not fully resolved. We then review the recent progress made on the transformed incidence problem, based on a new algebraic approach, originally introduced by Guth and Katz. Full details of the results reviewed
On the number of tetrahedra with minimum, unit, and distinct volumes in threespace ∗
, 710
"... We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by n points in 3space, and in general in d dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by n points in R3 is at most 2 3n3 − O(n2), and there are point sets for ..."
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We formulate and give partial answers to several combinatorial problems on volumes of simplices determined by n points in 3space, and in general in d dimensions. (i) The number of tetrahedra of minimum (nonzero) volume spanned by n points in R3 is at most 2 3n3 − O(n2), and there are point sets for which this number is 3 16n3 − O(n2). We also present an O(n3) time algorithm for reporting all tetrahedra of minimum nonzero volume, and thereby extend an algorithm of Edelsbrunner, O’Rourke, and Seidel. In general, for every k, d ∈ N, 1 ≤ k ≤ d, the maximum number of kdimensional simplices of minimum (nonzero) volume spanned by n points in R d is Θ(n k). (ii) The number of unitvolume tetrahedra determined by n points in R 3 is O(n 7/2), and there are point sets for which this number is Ω(n 3 log log n). (iii) For every d ∈ N, the minimum number of distinct volumes of all fulldimensional simplices determined by n points in R d, not all on a hyperplane, is Θ(n). 1
Collinearities in Kinetic Point Sets
, 2011
"... Let P be a set of n points in the plane, each point moving along a given trajectory. A kcollinearity is a pair (L, t) of a line L and a time t such that L contains at least k points at time t, L is spanned by the points at time t (i.e., the points along L are not all coincident), and not all of the ..."
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Let P be a set of n points in the plane, each point moving along a given trajectory. A kcollinearity is a pair (L, t) of a line L and a time t such that L contains at least k points at time t, L is spanned by the points at time t (i.e., the points along L are not all coincident), and not all of the points are collinear at all times. We show that, if the points move with constant velocity, then the number of 3collinearities is at most 2 ()
UNEXPECTED APPLICATIONS OF POLYNOMIALS IN COMBINATORICS
"... In the last six years, several combinatorics problems have been solved in an unexpected way using high degree polynomials. The most wellknown of these problems is the distinct distance problem in the plane. In [Erdős46], Erdős asked what is the smallest number of distinct distances determined by ..."
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In the last six years, several combinatorics problems have been solved in an unexpected way using high degree polynomials. The most wellknown of these problems is the distinct distance problem in the plane. In [Erdős46], Erdős asked what is the smallest number of distinct distances determined by n points in the plane. He noted that a square grid determines ∼ n(log n)−1/2 distinct distances, and he conjectured that this is sharp up to constant factors. Recently, an estimate was proven which is sharp up to logarithmic factors. Theorem 0.1. ([GuthKatz11], building on [ElekesSharir10]) For any n point set in the plane, the number of distinct distances is ≥ cn(log n)−1. The main new thing in the proof is the use of highdegree polynomials. This new technique first appeared in Dvir’s paper [Dvir09], which solved the finite field Nikodym and Kakeya problems. Experts had considered these problems very difficult, but the proof was essentially one page long. The method has had several other applications. The joints problem was resolved in [GuthKatz10]. The argument was simplified and generalized in [KSS10] and [Quilodrán10], leading to another one page proof. A higherdimensional generalization of the SzemerédiTrotter theorem was proven in [SolymosiTao12]. And