Results 1  10
of
50
A POLYNOMIAL BOUND IN FREIMAN’S THEOREM
 DUKE MATHEMATICAL JOURNAL VOL. 113, NO. 3
, 2002
"... In this paper the following improvement on Freiman’s theorem on set addition is obtained (see Theorems 1 and 2 in Section 1). Let A ⊂ Z be a finite set such that A + A  < αA. Then A is contained in a proper ddimensional progression P, where d ≤ [α − 1] and log(P/A) < Cα 2 (log α) 3. Earlier ..."
Abstract

Cited by 43 (2 self)
 Add to MetaCart
In this paper the following improvement on Freiman’s theorem on set addition is obtained (see Theorems 1 and 2 in Section 1). Let A ⊂ Z be a finite set such that A + A  < αA. Then A is contained in a proper ddimensional progression P, where d ≤ [α − 1] and log(P/A) < Cα 2 (log α) 3. Earlier bounds involved exponential dependence in α in the second estimate. Our argument combines I. Ruzsa’s method, which we improve in several places, as well as Y. Bilu’s proof of Freiman’s theorem.
On the Number of Incidences Between Points and Curves
 Combinatorics, Probability and Computing 7
"... We apply an idea of Sz'ekely to prove a general upper bound on the number of incidences between a set of m points and a set of n "wellbehaved" curves in the plane. ..."
Abstract

Cited by 28 (13 self)
 Add to MetaCart
We apply an idea of Sz'ekely to prove a general upper bound on the number of incidences between a set of m points and a set of n "wellbehaved" curves in the plane.
Applications of the crossing number
 In Proc. 10th Annu. ACM Sympos. Comput. Geom
, 1994
"... Abstract. The crossing number of a graph G is the minimum number of crossings in a drawing of G. The determination of the crossing number is an NPcomplete problem. We present two general lower bounds for the crossing number, and survey their applications and generalizations. 1 ..."
Abstract

Cited by 26 (6 self)
 Add to MetaCart
Abstract. The crossing number of a graph G is the minimum number of crossings in a drawing of G. The determination of the crossing number is an NPcomplete problem. We present two general lower bounds for the crossing number, and survey their applications and generalizations. 1
Improving the Crossing Lemma by Finding More Crossings in Sparse Graphs
, 2006
"... Twenty years ago, Ajtai et al. and, independently, Leighton discovered that the crossing number of any graph with v vertices and e> 4v edges is at least ce3 /v2, where c> 0 is an absolute constant. This result, known as the “Crossing Lemma, ” has found many important applications in discrete and com ..."
Abstract

Cited by 21 (1 self)
 Add to MetaCart
Twenty years ago, Ajtai et al. and, independently, Leighton discovered that the crossing number of any graph with v vertices and e> 4v edges is at least ce3 /v2, where c> 0 is an absolute constant. This result, known as the “Crossing Lemma, ” has found many important applications in discrete and computational geometry. It is tight up to a multiplicative constant. Here we improve the best known value of the constant by showing that the result holds with c> 1024/31827> 0.032. The proof has two new ingredients, interesting in their own right. We show that (1) if a graph can be drawn in the plane so that every edge crosses at most three others, then its number of edges cannot exceed 5.5(v − 2); and (2) the crossing number of any graph is at least 7 25 e − (v − 2). Both bounds are tight upt o
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. ..."
Abstract

Cited by 16 (0 self)
 Add to MetaCart
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.
On Distinct Sums and Distinct Distances
, 2001
"... The paper [10] of J. Solymosi and Cs. Toth implicitly raised the following arithmetic problem. Consider n pairwise disjoint s element sets and form all s 2 n sums of pairs of elements of the same set. What is the minimum number of distinct sums one can get this way? This paper proves a lower bo ..."
Abstract

Cited by 13 (3 self)
 Add to MetaCart
The paper [10] of J. Solymosi and Cs. Toth implicitly raised the following arithmetic problem. Consider n pairwise disjoint s element sets and form all s 2 n sums of pairs of elements of the same set. What is the minimum number of distinct sums one can get this way? This paper proves a lower bound on the number of distinct sums. As an application we improve the SolymosiToth bound on an old Erd}os problem: the number of distinct distances n points determine in the plane. Our bound also nds applications in other related results in discrete geometry. Our bounds are proven through an involved calculation of entropies of several random variables.
The Centroid of Points with Approximate Weights
 In Proc. 3rd Eur. Symp. Algorithms, number 979 in Lect. Notes in C. S
, 1995
"... . Let S be a set of points in IR d , each with a weight that is not known precisely, only known to fall within some range. What is the locus of the centroid of S? We prove that this locus is a convex polytope, the projection of a zonotope in IR d+1 . We derive complexity bounds and algorithm ..."
Abstract

Cited by 12 (4 self)
 Add to MetaCart
. Let S be a set of points in IR d , each with a weight that is not known precisely, only known to fall within some range. What is the locus of the centroid of S? We prove that this locus is a convex polytope, the projection of a zonotope in IR d+1 . We derive complexity bounds and algorithms for the construction of these "centroid polytopes". 1 Introduction Suppose that S = fs 1 ; s 2 ; : : : ; s n g is a set of points in IR d and that each point s i has an unknown nonnegative weight w i that lies within a known range [l i ; h i ]. The centroid of Salso called its "center of mass" or "weighted average"is the vector sum 1 W P i w i s i , where W = P i w i . We are interested in C(S), the set of all possible centroids. A generalization allows explicit upper and lower bounds on the total weight W , tighter than the implicit bounds P i l i W P i h i . Let C L;H (S) denote the locus of possible centroids of S with the additional constraint that L W H. T...
Counting Facets and Incidences
 DISCRETE COMPUT GEOM 7:359369 (1992)
, 1992
"... We show that m distinct cells in an arrangement of n planes in R a are bounded by O(m2/an + n 2) faces, which in turn yields a tight bound on the maximum number of facets bounding m cells in an arrangement of n hyperplanes in R a, for every d> 3. In addition, the method is extended to obtain tight b ..."
Abstract

Cited by 11 (3 self)
 Add to MetaCart
We show that m distinct cells in an arrangement of n planes in R a are bounded by O(m2/an + n 2) faces, which in turn yields a tight bound on the maximum number of facets bounding m cells in an arrangement of n hyperplanes in R a, for every d> 3. In addition, the method is extended to obtain tight bounds on the maximum number of faces on the boundary of all nonconvex cells in an arrangement of triangles in R 3. We also present a simpler proof of the O(m2/3n ~/3 + n d 1) bound on the number of incidences between n hyperplanes in R a and m vertices of their arrangement.
Distinct distances in three and higher dimensions
 Combin. Probab. Comput
, 2003
"... Improving an old result of Clarkson et al., we show that the number of distinct distances determined by a set P of n points in threedimensional space is Ω(n77/141−ε) = Ω(n0.546), for any ε> 0. Moreover, there always exists a point p ∈ P from which there are at least so many distinct distances to th ..."
Abstract

Cited by 11 (2 self)
 Add to MetaCart
Improving an old result of Clarkson et al., we show that the number of distinct distances determined by a set P of n points in threedimensional space is Ω(n77/141−ε) = Ω(n0.546), for any ε> 0. Moreover, there always exists a point p ∈ P from which there are at least so many distinct distances to the remaining elements of P. The same result holds for points on the threedimensional sphere. As a consequence, we obtain analogous results in higher dimensions. 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 ..."
Abstract

Cited by 11 (7 self)
 Add to MetaCart
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.