DOCSLIB.ORG

Rudnev, M. (2018). on the Number of Incidences Between Points and Planes in Three Dimensions

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Rudnev, M. (2018). on the Number of Incidences Between Points and Planes in Three Dimensions Rudnev, M. (2018). On the number of incidences between points and planes in three dimensions. Combinatorica, 38(1), 219–254. https://doi.org/10.1007/s00493-016-3329-6 Peer reviewed version License (if available): Unspecified Link to published version (if available): 10.1007/s00493-016-3329-6 Link to publication record in Explore Bristol Research PDF-document This is the accepted author manuscript (AAM). The final published version (version of record) is available online via Springer at http://link.springer.com/article/10.1007/s00493-016-3329-6#enumeration . Please refer to any applicable terms of use of the publisher. University of Bristol - Explore Bristol Research General rights This document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/red/research-policy/pure/user-guides/ebr-terms/ ON THE NUMBER OF INCIDENCES BETWEEN POINTS AND PLANES IN THREE DIMENSIONS MISHA RUDNEV Abstract. We prove an incidence theorem for points and planes in the projective space P3 over any field F, whose characteristic p = 2. An incidence is viewed as an intersection 6 along a line of a pair of two-planes from two canonical rulings of the Klein quadric. The Klein quadric can be traversed by a generic hyperplane, yielding a line-line incidence problem in a three-quadric, the Klein image of a regular line complex. This hyperplane can be chosen so that at most two lines meet. Hence, one can apply an algebraic theorem of Guth and Katz, with a constraint involving p if p> 0. This yields a bound on the number of incidences between m points and n planes in P3, with m n as ≥ O m√n + mk , where k is the maximum number of collinear planes, provided that n = O(p2) if p > 0. Examples show that this bound cannot be improved without additional assumptions. This gives one a vehicle to establish geometric incidence estimates when p > 0. For a non-collinear point set S F2 and a non-degenerate symmetric or skew-symmetric bilin- ⊆ 2 ear form ω, the number of distinct values of ω on pairs of points of S is Ω min S 3 ,p . h | | i This is also the best known bound over R, where it follows from the Szemer´edi-Trotter theorem. Also, a set S F3, not supported in a single semi-isotropic plane contains a ⊆ 1 point, from which Ω min S 2 ,p distinct distances to other points of S are attained. h | | i 1. Introduction Let F be a field of characteristic p and Pd the d-dimensional projective space over F. Our methods do not work for p = 2, but the results in view of constraints in terms of p hold trivially for p = O(1). As usual, we use the notation for cardinalities of finite sets. The symbols , , suppress absolute constants in inequalities,| · | as well as respectively do O and Ω. Besides,≪ X≫= Θ(Y ) means that X = O(Y ) and X = Ω(Y ). The symbols C and c stand for absolute constants, which may sometimes change from line to line, depending on the arXiv:1407.0426v5 [math.CO] 4 Dec 2015 context. When we turn to sum-products, we use the standard notation A + B = a + b : a A, b B { ∈ ∈ } for the sumset A + B of A, B F, similarly for the product set AB. ⊆ The Szemer´edi-Trotter theorem [37] on the number of incidences between lines and points in the Euclidean plane has many applications in combinatorics. The theorem is also valid over C, this was first proved by T´oth [38]. In positive characteristic, however, no universal satisfactory for applications point-line incidence estimate is available. The current “world record” for partial results in this direction for the prime residue field Fp is due to Jones [21]. This paper shows that in three dimensions there is an incidence estimate between a set P of m points and a set Π of n planes in P3, valid for any field of characteristic p = 2. If p> 0, there is a constraint that min(m,n)= O(p2). Hence, the result is trivial, unless6 p is 2000 Mathematics Subject Classification. 68R05,11B75. 1 2 MISHA RUDNEV regarded as a large parameter. Still, since our geometric set-up in terms of α-and β-planes in the Klein quadric breaks down for p = 2, we have chosen to state that p = 2 explicitly in the formulation of main results. Extending the results to more specific situations,6 when the constraint in terms of p can be weakened may not be impossible but way beyond the methodology herein. A few more words address this issue in the sequel. The set of incidences is defined as (1) I(P, Π) := (q, π) P Π: q π . { ∈ × ∈ } Over the reals, the point-plane incidence problem has been studied quite thoroughly throughout the past 25 years and several tight bounds are known. In general one can have all the points and planes involved to be incident to a single line in space, in which case the number of incidences is trivially mn. To do better than that, one needs some non- degeneracy assumption regarding collinearity, and the results quoted next differ as to the exact formulation of such an assumption. In the 1990 paper of Edelsbrunner et al. [10] it was proven (modulo slow-growing factors that can be removed, see [1]) that if no three planes are collinear in R3, 4 3 (2) I(P, Π) = O m 5 n 5 + m + n . | | This bound was shown to be tight for a wide range of m and n, owing to a construction by Brass and Knauer [7]. A thorough review of the state of the art by the year 2007 can be found in the paper of Apfelbaum and Sharir [1]. Elekes and T´oth [12] weakened the non-collinearity assumption down to that all planes were “not-too-degenerate”. That is a single line in a plane may support only a constant proportion of incidences in that plane. They proved a bound 3 (3) I(P, Π) = O (mn) 4 + m√n + n | | and presented a construction, showing it to be generally tight. The constructions supporting the tightness of both latter estimates are algebraic and extend beyond the real case. More recently, research in incidence geometry over R has intensified after the introduction of the polynomial partitioning technique in a breakthrough paper of Guth and Katz [14]. E.g., there is now a “continuous” generalisation of the bound (2) by Basit and Sheffer [3]: 4 +ǫ 3 2 (4) I(P, Π) = O∗ m 5 n 5 k 5 + mk + n , | | where k is the maximum number of collinear planes. For any ǫ> 0, the constant hidden in the O∗-symbol depends on ǫ. The proofs of the above results rely crucially on the order properties of R. Some of them, say (3) extend over C, for it is based on the Szemer´edi-Trotter theorem. Technically harder partitioning-based works like [3] have so far defied generalisation beyond R. This paper presents a different approach to point-plane incidences in the projective three- space P3. The approach appears to be robust enough to embrace, in principle, all fields F, but for the apparently special case of characteristic 2. When we have a specific field F in mind, we use the notation FP for the projective line P. The novelty of our approach is on its geometric side: we fetch and use extensively the classical XIX century Pl¨ucker- Klein formalism for line geometry in P3. This is combined with a recent algebraic incidence theorem for counting line-line intersections in three dimensions by Guth and Katz. The work of Guth and Katz, see [14] and the references contained therein for its prede- cessors, established two important theorems. Both rested on the polynomial Nullstellensatz ON THE NUMBER OF INCIDENCES BETWEEN POINTS AND PLANES IN THREE DIMENSIONS 3 principle, which was once again demonstrated to be so efficient a tool for discrete geometry problems by Dvir, who used it to resolve the finite field Kakeya conjecture [8]. The proof of the first Guth-Katz theorem, Theorem 2.10 in [14], was in essence algebraic, using the Nullstellensatz and basic properties of ruled surfaces, which come into play due to the use of the classical XIX century geometry Monge-Salmon theorem. See [33] for the original exposition of the latter theorem, as well as [23] (wholly dedicated to the prominent role this theorem plays in today’s incidence geometry) and Appendix in [24]. The second Guth-Katz theorem, Theorem 2.11 in [14], introduced the aforementioned method of polynomial partitioning of the real space, based on the Borsuk-Ulam theorem. It is the latter theorem of Guth and Katz that has recently attracted more attention and follow-ups. Since we work over any field, we cannot not use polynomial partitioning. It is a variant of Theorem 2.10 from [14] that plays a key role here, and is henceforth referred to as the Guth-Katz theorem. We share this, at least in part, with a recent work of Koll´ar [24] dedicated to point-line incidences in 3D, in particular over fields with positive characteristic. Theorem 1 (Guth-Katz). Let be a set of n straight lines in R3. Suppose, no more then two lines are concurrent. ThenL the number of pair-wise intersections of lines in is L 3 O n 2 + nk , where k is the maximum number of lines, contained in a plane or regulus.
Load more

Recommended publications
  • Study Guide for the Midterm. Topics: • Euclid • Incidence Geometry • Perspective Geometry • Neutral Geometry – Through Oct.18Th
    Study guide for the midterm. Topics: • Euclid • Incidence geometry • Perspective geometry • Neutral geometry { through Oct.18th. You need to { Know definitions and examples. { Know theorems by names. { Be able to give a simple proof and justify every step. { Be able to explain those of Euclid's statements and proofs that appeared in the book, the problem set, the lecture, or the tutorial. Sources: • Class Notes • Problem Sets • Assigned reading • Notes on course website. The test. • Excerpts from Euclid will be provided if needed. • The incidence axioms will be provided if needed. • Postulates of neutral geometry will be provided if needed. • There will be at least one question that is similar or identical to a homework question. Tentative large list of terms. For each of the following terms, you should be able to write two sentences, in which you define it, state it, explain it, or discuss it. Euclid's geometry: - Euclid's definition of \right angle". - Euclid's definition of \parallel lines". - Euclid's \postulates" versus \common notions". - Euclid's working with geometric magnitudes rather than real numbers. - Congruence of segments and angles. - \All right angles are equal". - Euclid's fifth postulate. - \The whole is bigger than the part". - \Collapsing compass". - Euclid's reliance on diagrams. - SAS; SSS; ASA; AAS. - Base angles of an isosceles triangle. - Angles below the base of an isosceles triangle. - Exterior angle inequality. - Alternate interior angle theorem and its converse (we use the textbook's convention). - Vertical angle theorem. - The triangle inequality. - Pythagorean theorem. Incidence geometry: - collinear points. - concurrent lines. - Simple theorems of incidence geometry. - Interpretation/model for a theory.
    [Show full text]
  • Arxiv:1609.06355V1 [Cs.CC] 20 Sep 2016 Outlaw Distributions And
    Outlaw distributions and locally decodable codes Jop Bri¨et∗ Zeev Dvir† Sivakanth Gopi‡ September 22, 2016 Abstract Locally decodable codes (LDCs) are error correcting codes that allow for decoding of a single message bit using a small number of queries to a corrupted encoding. De- spite decades of study, the optimal trade-off between query complexity and codeword length is far from understood. In this work, we give a new characterization of LDCs using distributions over Boolean functions whose expectation is hard to approximate (in L∞ norm) with a small number of samples. We coin the term ‘outlaw distribu- tions’ for such distributions since they ‘defy’ the Law of Large Numbers. We show that the existence of outlaw distributions over sufficiently ‘smooth’ functions implies the existence of constant query LDCs and vice versa. We also give several candidates for outlaw distributions over smooth functions coming from finite field incidence geometry and from hypergraph (non)expanders. We also prove a useful lemma showing that (smooth) LDCs which are only required to work on average over a random message and a random message index can be turned into true LDCs at the cost of only constant factors in the parameters. 1 Introduction Error correcting codes (ECCs) solve the basic problem of communication over noisy channels. They encode a message into a codeword from which, even if the channel partially corrupts it, arXiv:1609.06355v1 [cs.CC] 20 Sep 2016 the message can later be retrieved. With one of the earliest applications of the probabilistic method, formally introduced by Erd˝os in 1947, pioneering work of Shannon [Sha48] showed the existence of optimal (capacity-achieving) ECCs.
    [Show full text]
  • Connections Between Graph Theory, Additive Combinatorics, and Finite
    UNIVERSITY OF CALIFORNIA, SAN DIEGO Connections between graph theory, additive combinatorics, and finite incidence geometry A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy in Mathematics by Michael Tait Committee in charge: Professor Jacques Verstra¨ete,Chair Professor Fan Chung Graham Professor Ronald Graham Professor Shachar Lovett Professor Brendon Rhoades 2016 Copyright Michael Tait, 2016 All rights reserved. The dissertation of Michael Tait is approved, and it is acceptable in quality and form for publication on microfilm and electronically: Chair University of California, San Diego 2016 iii DEDICATION To Lexi. iv TABLE OF CONTENTS Signature Page . iii Dedication . iv Table of Contents . .v List of Figures . vii Acknowledgements . viii Vita........................................x Abstract of the Dissertation . xi 1 Introduction . .1 1.1 Polarity graphs and the Tur´annumber for C4 ......2 1.2 Sidon sets and sum-product estimates . .3 1.3 Subplanes of projective planes . .4 1.4 Frequently used notation . .5 2 Quadrilateral-free graphs . .7 2.1 Introduction . .7 2.2 Preliminaries . .9 2.3 Proof of Theorem 2.1.1 and Corollary 2.1.2 . 11 2.4 Concluding remarks . 14 3 Coloring ERq ........................... 16 3.1 Introduction . 16 3.2 Proof of Theorem 3.1.7 . 21 3.3 Proof of Theorems 3.1.2 and 3.1.3 . 23 3.3.1 q a square . 24 3.3.2 q not a square . 26 3.4 Proof of Theorem 3.1.8 . 34 3.5 Concluding remarks on coloring ERq ........... 36 4 Chromatic and Independence Numbers of General Polarity Graphs .
    [Show full text]
  • Keith E. Mellinger
    Keith E. Mellinger Department of Mathematics home University of Mary Washington 1412 Kenmore Avenue 1301 College Avenue, Trinkle Hall Fredericksburg, VA 22401 Fredericksburg, VA 22401-5300 (540) 368-3128 ph: (540) 654-1333, fax: (540) 654-2445 [email protected] http://www.keithmellinger.com/ Professional Experience • Director of the Quality Enhancement Plan, 2013 { present. UMW. • Professor of Mathematics, 2014 { present. Department of Mathematics, UMW. • Interim Director, Office of Academic and Career Services, Fall 2013 { Spring 2014. UMW. • Associate Professor, 2008 { 2014. Department of Mathematics, UMW. • Department Chair, 2008 { 2013. Department of Mathematics, UMW. • Assistant Professor, 2003 { 2008. Department of Mathematics, UMW. • Research Assistant Professor (Vigre post-doc), 2001 - 2003. Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago, Chicago, IL, 60607-7045. Position partially funded by a VIGRE grant from the National Science Foundation. Education • Ph.D. in Mathematics: August 2001, University of Delaware, Newark, DE. { Thesis: Mixed Partitions and Spreads of Projective Spaces • M.S. in Mathematics: May 1997, University of Delaware. • B.S. in Mathematics: May 1995, Millersville University, Millersville, PA. { minor in computer science, option in statistics Grants and Awards • Millersville University Outstanding Young Alumni Achievement Award, presented by the alumni association at MU, May 2013. • Mathematical Association of America's Carl B. Allendoerfer Award for the article,
    [Show full text]
  • 1 Definition and Models of Incidence Geometry
    1 Definition and Models of Incidence Geometry (1.1) Definition (geometry) A geometry is a set S of points and a set L of lines together with relationships between the points and lines. p 2 1. Find four points which belong to set M = f(x;y) 2 R jx = 5g: 2. Let M = f(x;y)jx;y 2 R;x < 0g: p (i) Find three points which belong to set N = f(x;y) 2 Mjx = 2g. − 4 L f 2 Mj 1 g (ii) Are points P (3;3), Q(6;4) and R( 2; 3 ) belong to set = (x;y) y = 3 x + 2 ? (1.2) Definition (Cartesian Plane) L Let E be the set of all vertical and non-vertical lines, La and Lk;n where vertical lines: f 2 2 j g La = (x;y) R x = a; a is fixed real number ; non-vertical lines: f 2 2 j g Lk;n = (x;y) R y = kx + n; k and n are fixed real numbers : C f 2 L g The model = R ; E is called the Cartesian Plane. C f 2 L g 3. Let = R ; E denote Cartesian Plane. (i) Find three different points which belong to Cartesian vertical line L7. (ii) Find three different points which belong to Cartesian non-vertical line L15;p2. C f 2 L g 4. Let P be some point in Cartesian Plane = R ; E . Show that point P cannot lie simultaneously on both L and L (where a a ). a a0 , 0 (1.3) Definition (Poincar´ePlane) L Let H be the set of all type I and type II lines, aL and pLr where type I lines: f 2 j g aL = (x;y) H x = a; a is a fixed real number ; type II lines: f 2 j − 2 2 2 2 g pLr = (x;y) H (x p) + y = r ; p and r are fixed R, r > 0 ; 2 H = f(x;y) 2 R jy > 0g: H f L g The model = H; H will be called the Poincar´ePlane.
    [Show full text]
  • ON the POLYHEDRAL GEOMETRY of T–DESIGNS
    ON THE POLYHEDRAL GEOMETRY OF t{DESIGNS A thesis presented to the faculty of San Francisco State University In partial fulfilment of The Requirements for The Degree Master of Arts In Mathematics by Steven Collazos San Francisco, California August 2013 Copyright by Steven Collazos 2013 CERTIFICATION OF APPROVAL I certify that I have read ON THE POLYHEDRAL GEOMETRY OF t{DESIGNS by Steven Collazos and that in my opinion this work meets the criteria for approving a thesis submitted in partial fulfillment of the requirements for the degree: Master of Arts in Mathematics at San Francisco State University. Matthias Beck Associate Professor of Mathematics Felix Breuer Federico Ardila Associate Professor of Mathematics ON THE POLYHEDRAL GEOMETRY OF t{DESIGNS Steven Collazos San Francisco State University 2013 Lisonek (2007) proved that the number of isomorphism types of t−(v; k; λ) designs, for fixed t, v, and k, is quasi{polynomial in λ. We attempt to describe a region in connection with this result. Specifically, we attempt to find a region F of Rd with d the following property: For every x 2 R , we have that jF \ Gxj = 1, where Gx denotes the G{orbit of x under the action of G. As an application, we argue that our construction could help lead to a new combinatorial reciprocity theorem for the quasi{polynomial counting isomorphism types of t − (v; k; λ) designs. I certify that the Abstract is a correct representation of the content of this thesis. Chair, Thesis Committee Date ACKNOWLEDGMENTS I thank my advisors, Dr. Matthias Beck and Dr.
    [Show full text]
  • Matroid Enumeration for Incidence Geometry
    Discrete Comput Geom (2012) 47:17–43 DOI 10.1007/s00454-011-9388-y Matroid Enumeration for Incidence Geometry Yoshitake Matsumoto · Sonoko Moriyama · Hiroshi Imai · David Bremner Received: 30 August 2009 / Revised: 25 October 2011 / Accepted: 4 November 2011 / Published online: 30 November 2011 © Springer Science+Business Media, LLC 2011 Abstract Matroids are combinatorial abstractions for point configurations and hy- perplane arrangements, which are fundamental objects in discrete geometry. Matroids merely encode incidence information of geometric configurations such as collinear- ity or coplanarity, but they are still enough to describe many problems in discrete geometry, which are called incidence problems. We investigate two kinds of inci- dence problem, the points–lines–planes conjecture and the so-called Sylvester–Gallai type problems derived from the Sylvester–Gallai theorem, by developing a new algo- rithm for the enumeration of non-isomorphic matroids. We confirm the conjectures of Welsh–Seymour on ≤11 points in R3 and that of Motzkin on ≤12 lines in R2, extend- ing previous results. With respect to matroids, this algorithm succeeds to enumerate a complete list of the isomorph-free rank 4 matroids on 10 elements. When geometric configurations corresponding to specific matroids are of interest in some incidence problems, they should be analyzed on oriented matroids. Using an encoding of ori- ented matroid axioms as a boolean satisfiability (SAT) problem, we also enumerate oriented matroids from the matroids of rank 3 on n ≤ 12 elements and rank 4 on n ≤ 9 elements. We further list several new minimal non-orientable matroids. Y. Matsumoto · H. Imai Graduate School of Information Science and Technology, University of Tokyo, Tokyo, Japan Y.
    [Show full text]
  • Set Theory. • Sets Have Elements, Written X ∈ X, and Subsets, Written a ⊆ X. • the Empty Set ∅ Has No Elements
    Set theory. • Sets have elements, written x 2 X, and subsets, written A ⊆ X. • The empty set ? has no elements. • A function f : X ! Y takes an element x 2 X and returns an element f(x) 2 Y . The set X is its domain, and Y is its codomain. Every set X has an identity function idX defined by idX (x) = x. • The composite of functions f : X ! Y and g : Y ! Z is the function g ◦ f defined by (g ◦ f)(x) = g(f(x)). • The function f : X ! Y is a injective or an injection if, for every y 2 Y , there is at most one x 2 X such that f(x) = y (resp. surjective, surjection, at least; bijective, bijection, exactly). In other words, f is a bijection if and only if it is both injective and surjective. Being bijective is equivalent to the existence of an inverse function f −1 such −1 −1 that f ◦ f = idY and f ◦ f = idX . • An equivalence relation on a set X is a relation ∼ such that { (reflexivity) for every x 2 X, x ∼ x; { (symmetry) for every x; y 2 X, if x ∼ y, then y ∼ x; and { (transitivity) for every x; y; z 2 X, if x ∼ y and y ∼ z, then x ∼ z. The equivalence class of x 2 X under the equivalence relation ∼ is [x] = fy 2 X : x ∼ yg: The distinct equivalence classes form a partition of X, i.e., every element of X is con- tained in a unique equivalence class. Incidence geometry.
    [Show full text]
  • Math 431 Solutions to Homework 3
    Math 431 Solutions to Homework 3 1. Show that the following familiar interpretation is a model for incidence geometry. Note that in class we already showed that Axiom I-1 is satisfied, so you only need to show that I-2 and I-3 are satisfied. • The “points” are all ordered pairs (x, y) of real numbers. • A “line” is specified by an ordered triple (a, b, c) of real numbers such that either a 6= 0 or b 6= 0; it is defined as the set of all “points” (x, y) that satisfy the equation ax + by + c = 0. • “Incidence” is defined as set membership. Proof that I-1 is satisfied. This is an expanded version of what was done in class. Let P = (x1, y1) and Q = (x2, y2) be two distinct points. We need to show that there is a unique line passing through both of them. Case 1: Suppose that x1 = x2. In this case the line given by the equation x = x1 (1) passes through P and Q (since (x1, y1) and (x2, y2) satisfy this equation). To show that this line is unique, suppose we are given any line l passing through P and Q. Let ax + by + c = 0 be an equation for l. Since P and Q satisfy this equation, ax1 + by1 + c = 0, and ax2 + by2 + c = 0. Now we have ax1 + by1 + c = ax2 + by2 + c ⇐⇒ ax1 + by1 = ax2 + by2 ⇐⇒ ax1 − ax2 = by2 − by1 ⇐⇒ −a(x2 − x1) = b(y2 − y1). We know that y1 6= y2 (because x1 = x2 and P 6= Q).
    [Show full text]
  • Analogy Between Additive Combinations, Finite Incidence Geometry & Graph Theory
    International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056 Volume: 04 Issue: 05 | May -2017 www.irjet.net p-ISSN: 2395-0072 Analogy between Additive Combinations, Finite incidence Geometry & Graph Theory 1 2 K SrnivasaRao , Dr k.Rameshbabu 1Associate Professor, Mathematics, H&S.CIET, JNTUK, Guntur, AP, India. 2Associate. Professor, Comm. stream, ECE Department, JIT, JU, East.Afrika --------------------------------------------------------------***-------------------------------------------------------------- Abstract: The topic is the study of the Tur´an number finite subsets of a group with the property that they are for C4. Fu¨redi showed that C4-free graphs with ex(n,C4) almost closed under multiplication. Approximate groups edges are intimately related to polarity graphs of and their applications (for example, to expand er graphs, projective planes. Then prove a general theorem about group theory, Probability, model theory, and so on) form dense subgraphs in a wide class of polarity graphs, and as a very active and promising area of research in additive a result give the best-known lower bounds for ex(n,C4) for combinatorics. this paper describes additive many values of n.and also study the chromatic and combinatorics as the following: “additive combinatorics independence numbers of polarity graphs, with special focuses on three classes of theorems: decomposition emphasis on the graph ERq. Next study is Sidon sets on theorems, approximate structural theorems, and graphs by considering what sets of integers may look like transference principles ”.Techniques and approaches when certain pairs of them are restricted from having the applied in additive combinatorics are often extremely same product. Other generalizations of Sidon sets are sophisticated, and may have roots in several unexpected considered as well.
    [Show full text]
  • Extending Erd\H {O} S-Beck's Theorem to Higher Dimensions
    EXTENDING ERDOS-˝ BECK’S THEOREM TO HIGHER DIMENSIONS THAO DO1 ABSTRACT. Erd˝os-Beck theorem states that n points in the plane with at most n − x points collinear define at least cxn lines for some positive constant c. It implies n points in the plane define Θ(n2) lines unless most of the points (i.e. n − o(n) points) are collinear. In this paper, we will present two ways to extend this result to higher dimensions. Given a set S of n points in Rd, we want to estimate a lower bound of the number of hyperplanes they define (a hyperplane is defined or spanned by S if it contains d +1 points of S in general position). Our first result says the number of spanned hyperplanes is at least cxnd−1 if there exists some hyperplane that contains n − x points of S and saturated (as defined in Definition 1.3). Our second result says n points in Rd define Θ(nd) hyperplanes unless most of the points belong to the union of a collection of flats whose sum of dimension is strictly less than d. Our result has application to point-hyperplane incidences and potential application to the point covering problem. 1. INTRODUCTION Given a set S of n points in the plane, we say a line l is a spanning line of S (or l is spanned by S) if l contains at least two distinct points of S. The following theorem was proposed by Erd˝os and proved by Beck in [3]: Theorem 1.1.
    [Show full text]
  • Structure Theorems and Extremal Problems in Incidence Geometry
    Structure theorems and extremal problems in incidence geometry Hiu Chung Aaron Lin A thesis submitted for the degree of Doctor of Philosophy Department of Mathematics The London School of Economics and Political Science November 2019 Declaration I certify that the thesis I have presented for examination for the MPhil/PhD degree of the London School of Economics and Political Science is my own work, with the following exceptions. Parts of Section 3.3, parts of Chapter4, Section 5.3, and Section 6.3 are based on [40], which is published in Discrete & Computational Geometry, and is joint work with Mehdi Makhul, Hossein Nassajian Mojarrad, Josef Schicho, Konrad Swanepoel, and Frank de Zeeuw. Lemma 3.16 of Section 3.2, Section 5.1, and Section 6.1 are based on [41], which is accepted to the Journal of the London Mathematical Society, and is joint work with Konrad Swanepoel. Lemma 2.6 of Section 2.1, Section 2.2.2, Section 3.2, parts of Chapter4, Section 5.2, and Section 6.2 are based on [42], which is joint work with Konrad Swanepoel. Parts of Section 3.3, Section 5.4, and Section 6.4 are based on [43], which is joint work with Konrad Swanepoel. The copyright of this thesis rests with the author. Quotation from it is permitted, provided that full acknowledgement is made. This thesis may not be reproduced without the prior written consent of the author. I warrant that this authorisation does not, to the best of my belief, infringe the rights of any third party.
    [Show full text]