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A sumproduct estimate in finite fields, and applications
"... Abstract. Let A be a subset of a finite field F: = Z/qZ for some prime q. If F  δ < A  < F  1−δ for some δ> 0, then we prove the estimate A + A  + A · A  ≥ c(δ)A  1+ε for some ε = ε(δ)> 0. This is a finite field analogue of a result of [ESz1983]. We then use this estimate to prove a Sze ..."
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Abstract. Let A be a subset of a finite field F: = Z/qZ for some prime q. If F  δ < A  < F  1−δ for some δ> 0, then we prove the estimate A + A  + A · A  ≥ c(δ)A  1+ε for some ε = ε(δ)> 0. This is a finite field analogue of a result of [ESz1983]. We then use this estimate to prove a SzemerédiTrotter type theorem in finite fields, and obtain a new estimate for the Erdös distance problem in finite fields, as well as the threedimensional Kakeya problem in finite fields. 1.
On Distinct Distances from a Vertex of a Convex Polygon
"... Given a set P of n points in convex position in the plane, we prove that there exists a point p 2 P such that the number of distinct distances from p is at least d(13n  6)/36e. The best previous bound, dn=3e, from 1952, is due to Leo Moser. ..."
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Given a set P of n points in convex position in the plane, we prove that there exists a point p 2 P such that the number of distinct distances from p is at least d(13n  6)/36e. The best previous bound, dn=3e, from 1952, is due to Leo Moser.
Fourier analysis and expanding phenomena in finite fields
 Proc. Amer. Math. Soc
"... Abstract. In this paper the authors study set expansion in finite fields. Fourier analytic proofs are given for several results recently obtained by Solymosi ([33]), Vu ([41]) and Vinh ([40]) using spectral graph theory. In addition, several generalizations of these results are given. In the case th ..."
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Cited by 2 (0 self)
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Abstract. In this paper the authors study set expansion in finite fields. Fourier analytic proofs are given for several results recently obtained by Solymosi ([33]), Vu ([41]) and Vinh ([40]) using spectral graph theory. In addition, several generalizations of these results are given. In the case that A is a subset of a prime field Fp of size less than p 1/2 it is shown that {a 2 + b: a, b ∈ A}  ≥ CA  147/146, where  ·  denotes the cardinality of the set and C is an absolute constant. 1. introduction Let F be a field and E be a finite subset of F d, the ddimensional vector space over F. Given a function f: F d → F define f(E) = {f(x) : x ∈ E}, the image of f under the subset E. We shall say that f is a dvariable expander with expansion index ǫ if f(E)  ≥ CǫE  1/d+ǫ for every subset E possibly under some general density or structural assumptions on E. Several classical problems in additive and geometric combinatorics deal with showing that certain polynomials have the expander property. Given a finite subset E ⊂ R d the Erdős distance problem deals with the case of ∆ : R d × R d → R where ∆(x, y) = ‖x − y‖. It is conjectured that ∆(E, E)  � E  2/d, that is ∆ is a 2dvariable expander with expansion index 1/2d. Taking E to be a piece of the integer lattice shows that one cannot in general do better. (Throughout the paper we will write X � Y to mean X ≤ CY where C is a universal constant, which may vary from line to line but are always universal. It is also clear that when the quantities X, Y have f(A) involved for some polynomial f, the implied constant may also depend on the degree of f. In addition, we will write X � Y in the case that for every δ> 0 there exists Cδ> 0 such that X ≤ Cδt δ Y where t is a large controlling parameter.)
A Bipartite Strengthening of the Crossing Lemma
"... Abstract. The celebrated Crossing Lemma states that, in every drawing of a graph with n vertices and m ≥ 4n edges there are at least Ω(m 3 /n 2) pairs of crossing edges; or equivalently, there is an edge that crosses Ω(m 2 /n 2) other edges. We strengthen the Crossing Lemma for drawings in which any ..."
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Abstract. The celebrated Crossing Lemma states that, in every drawing of a graph with n vertices and m ≥ 4n edges there are at least Ω(m 3 /n 2) pairs of crossing edges; or equivalently, there is an edge that crosses Ω(m 2 /n 2) other edges. We strengthen the Crossing Lemma for drawings in which any two edges cross in at most O(1) points. We prove for every k ∈ N that every graph G with n vertices and m ≥ 3n edges drawn in the plane such that any two edges intersect in at most k points has two disjoint subsets of edges, E1 and E2, each of size at least ckm 2 /n 2, such that every edge in E1 crosses all edges in E2, where ck> 0 only depends on k. This bound is best possible up to the constant ck for every k ∈ N. We also prove that every graph G with n vertices and m ≥ 3n edges drawn in the plane with xmonotone edges has disjoint subsets of edges, E1 and E2, each of size Ω(m 2 /(n 2 polylog n)), such that every edge in E1 crosses all edges in E2. On the other hand, we construct xmonotone drawings of bipartite dense graphs where the largest such subsets E1 and E2 have size O(m 2 /(n 2 log(m/n))). 1
DISTINCT DISTANCES IN GRAPH DRAWINGS
, 2008
"... The distancenumber of a graph G is the minimum number of distinct edgelengths over all straightline drawings of G in the plane. This definition generalises many wellknown concepts in combinatorial geometry. We consider the distancenumber of trees, graphs with no K − 4minor, complete bipartite g ..."
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The distancenumber of a graph G is the minimum number of distinct edgelengths over all straightline drawings of G in the plane. This definition generalises many wellknown concepts in combinatorial geometry. We consider the distancenumber of trees, graphs with no K − 4minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distancenumber of graphs with bounded degree. We prove that nvertex graphs with bounded maximum degree and bounded treewidth have distancenumber in O(log n). To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth 2 and polynomial distancenumber. Similarly, we prove that there exist graphs with maximum degree 5 and arbitrarily large distancenumber. Moreover, as ∆ increases the existential lower bound on the distancenumber of ∆regular graphs tends to Ω(n0.864138). 1
LECTURE NOTES 1 FOR 254A
"... The aim of this course is to tour the highlights of arithmetic combinatorics the combinatorial estimates relating to the sums, differences, and products of finite sets, or to related objects such as arithmetic progressions. The material here is of course mostly combinatorial, but we will also explo ..."
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The aim of this course is to tour the highlights of arithmetic combinatorics the combinatorial estimates relating to the sums, differences, and products of finite sets, or to related objects such as arithmetic progressions. The material here is of course mostly combinatorial, but we will also exploit the Fourier transform at times. We
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"... 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 ∗
The Beginnings of Geometric Graph Theory
"... “...to ask the right question and to ask it of the right person.” (Richard Guy) Geometric graphs (topological graphs) are graphs drawn in the plane with possibly crossing straightline edges (resp., curvilinear edges). Starting with a problem of Heinz Hopf and Erika Pannwitz from 1934 and a seminal ..."
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“...to ask the right question and to ask it of the right person.” (Richard Guy) Geometric graphs (topological graphs) are graphs drawn in the plane with possibly crossing straightline edges (resp., curvilinear edges). Starting with a problem of Heinz Hopf and Erika Pannwitz from 1934 and a seminal paper of Paul Erdős from 1946, we give a biased survey of Turántype questions in the theory of geometric and topological graphs. What is the maximum number of edges that a geometric or topological graph of n vertices can have if it contains no forbidden subconfiguration of a certain type? We put special emphasis on open problems raised by Erdős or directly motivated by his work. 1