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Crossing numbers and hard Erdős problems in discrete geometry
 COMBINATORICS, PROBABILITY AND COMPUTING
, 1997
"... We show that an old but not wellknown lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the min ..."
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Cited by 113 (1 self)
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We show that an old but not wellknown lower bound for the crossing number of a graph yields short proofs for a number of bounds in discrete plane geometry which were considered hard before: the number of incidences among points and lines, the maximum number of unit distances among n points, the minimum number of distinct distances among n points.
Distinct Distances in the Plane
, 2001
"... It is shown that every set of n points in the plane has an element from which there are at least cn 6/7 other elements at distinct distances, where c > 0 is a constant. This improves earlier results of Erdos, Moser, Beck, Chung, Szemeredi, Trotter, and Szekely. 1. ..."
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Cited by 16 (0 self)
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It is shown that every set of n points in the plane has an element from which there are at least cn 6/7 other elements at distinct distances, where c > 0 is a constant. This improves earlier results of Erdos, Moser, Beck, Chung, Szemeredi, Trotter, and Szekely. 1.
Some connections between Falconer’s distance set conjecture, and sets of Furstenburg type
, 2001
"... ..."
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 10 (5 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.
Vol. 102, No. 2 DUKE MATHEMATICAL JOURNAL © 2000 SMOOTHNESS OF PROJECTIONS, BERNOULLI CONVOLUTIONS, AND THE DIMENSION OF EXCEPTIONS
"... 0 ±λ n, where the signs are chosen independently with probability 1 2, has been studied by many authors since the two seminal papers by Erdős in 1939 and 1940. It is immediate that νλ ..."
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0 ±λ n, where the signs are chosen independently with probability 1 2, has been studied by many authors since the two seminal papers by Erdős in 1939 and 1940. It is immediate that νλ
The Exact Fitting . . .
 COMPUT. GEOM. THEORY APPL
, 1992
"... Let S be a family of n points in E a. The exact fitting problem asks for finding a hyperplane containing the maximum number of points of $. In this paper, we present an O (rain j' " " }) ,aha log , n d time algorithm where m denoted the number of points the hyperplane. This algorithm is bas ..."
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Let S be a family of n points in E a. The exact fitting problem asks for finding a hyperplane containing the maximum number of points of $. In this paper, we present an O (rain j' " " }) ,aha log , n d time algorithm where m denoted the number of points the hyperplane. This algorithm is based on upper bounds on the maximum number of incidences between families of points and families ofhyperplanes in E d and on an algorithm to compute these incidences. We also show how the upper bound on the maximum number of incidences between families of points and families of hyperplanes can be used to derive new bounds on some wellknown problems in discrete geometry.
New
"... results on the distribution of distances determined by separated point sets ∗ ..."
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results on the distribution of distances determined by separated point sets ∗
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