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11
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 109 (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.
SMOOTHNESS OF PROJECTIONS, BERNOULLI CONVOLUTIONS, AND THE DIMENSION OF EXCEPTIONS
 VOL. 102, NO. 2 DUKE MATHEMATICAL JOURNAL
, 2000
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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
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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 11 (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.
The number of distinct distances from a vertex of a convex polygon
 J. Comput. Geom
"... Erdős conjectured in 1946 that every npoint set P in convex position in the plane has a point that determines at least ⌊n/2 ⌋ distinct distances to the other points of P. In 2006 Dumitrescu improved the best known lower bound for this problem, from n/3 to 13n/36 − O(1). A crucial step in his argume ..."
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Cited by 1 (0 self)
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Erdős conjectured in 1946 that every npoint set P in convex position in the plane has a point that determines at least ⌊n/2 ⌋ distinct distances to the other points of P. In 2006 Dumitrescu improved the best known lower bound for this problem, from n/3 to 13n/36 − O(1). A crucial step in his argument is showing that P must determine at most n2 (1 − 1/12) isosceles triangles. In this paper we show that Dumitrescu’s bound can be further improved, though our improvement is quite small. We show that the number of isosceles triangles determined by P is at most n2 (1 − 1/11.981), and we conclude that there exists a point of P that determines at least
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.
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