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Incidence Theorems and Their Applications
"... We survey recent (and not so recent) results concerning arrangements of lines, points and other geometric objects and the applications these results have in theoretical computer science and combinatorics. The three main types of problems we will discuss are: 1. Counting incidences: Given a set (or s ..."
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We survey recent (and not so recent) results concerning arrangements of lines, points and other geometric objects and the applications these results have in theoretical computer science and combinatorics. The three main types of problems we will discuss are: 1. Counting incidences: Given a set (or several sets) of geometric objects (lines, points, etc.), what is the maximum number of incidences (or intersections) that can exist between elements in different sets? We will see several results of this type, such as the SzemerediTrotter theorem, over the reals and over finite fields and discuss their applications in combinatorics (e.g., in the recent solution of Guth and Katz to Erdos ’ distance problem) and in computer science (in explicit constructions of multisource extractors). 2. Kakeya type problems: These problems deal with arrangements of lines that point in different directions. The goal is to try and understand to what extent these lines can overlap one another. We will discuss these questions both over the reals and over finite fields and see how they come up in the
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
Almost Tight Bounds for Eliminating Depth Cycles in Three Dimensions∗
, 2015
"... Given n nonvertical lines in 3space, their vertical depth (above/below) relation can contain cycles. We show that the lines can be cut into O(n3/2 polylog n) pieces, such that the depth relation among these pieces is now a proper partial order. This bound is nearly tight in the worst case. As a co ..."
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Given n nonvertical lines in 3space, their vertical depth (above/below) relation can contain cycles. We show that the lines can be cut into O(n3/2 polylog n) pieces, such that the depth relation among these pieces is now a proper partial order. This bound is nearly tight in the worst case. As a consequence, we deduce that the number of pairwise nonoverlapping cycles, namely, cycles whose xyprojections do not overlap, is O(n3/2 polylog n); this bound too is almost tight in the worst case. Previous results on this topic could only handle restricted cases of the problem (such as handling only triangular cycles, by Aronov, Koltun, and Sharir, or only cycles in gridlike patterns, by Chazelle et al.), and the bounds were considerably weaker—much closer to quadratic. Our proof uses a recent variant of the polynomial partitioning technique, due to Guth, and some simple tools from algebraic geometry. It is much more straightforward than the previous “purely combinatorial ” methods. Our technique extends to eliminating all cycles in the depth relation among segments, and of constantdegree algebraic arcs. We hope that a suitable extension of this technique could be used to handle the (much more difficult) case of pairwisedisjoint triangles. Our results almost completely settle a longstanding (35 years old) open problem in computational geometry, motivated by hiddensurface removal in computer graphics.
THE JOINTS PROBLEM IN n
, 906
"... n Abstract. We show that given a collection of A lines in, n � 2, the maximum number of their joints (points incident to at least n lines whose directions form a linearly independent set) is O(An/(n−1)). An analogous result for smooth algebraic curves is also proven. 1. ..."
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n Abstract. We show that given a collection of A lines in, n � 2, the maximum number of their joints (points incident to at least n lines whose directions form a linearly independent set) is O(An/(n−1)). An analogous result for smooth algebraic curves is also proven. 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 ()
Discrete and Computational Geometry manuscript No. (will be inserted by the editor) Generalizations of the SzemerédiTrotter Theorem
"... Abstract We generalize the SzemerédiTrotter incidence theorem, to bound the number of complete flags in higher dimensions. Specifically, for each i = 0, 1,..., d − 1, we are given a finite set Si of iflats in Rd or in Cd, and a (complete) flag is a tuple (f0, f1,..., fd−1), where fi ∈ Si for each ..."
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Abstract We generalize the SzemerédiTrotter incidence theorem, to bound the number of complete flags in higher dimensions. Specifically, for each i = 0, 1,..., d − 1, we are given a finite set Si of iflats in Rd or in Cd, and a (complete) flag is a tuple (f0, f1,..., fd−1), where fi ∈ Si for each i and fi ⊂ fi+1 for each i = 0, 1,..., d − 2. Our main result is an upper bound on the number of flags which is tight in the worst case. We also study several other kinds of incidence problems, including (i) incidences between points and lines in R3 such that among the lines incident to a point, at most O(1) of them can be coplanar, (ii) incidences with Legendrian lines in R3, a special class of lines that arise when considering flags that are defined in terms of other groups, and (iii) flags in R3 (involving points, lines, and planes), where no given line can contain too many points or lie on too many planes. The bound that we obtain in (iii) is nearly tight in the worst case.