TY - JOUR

T1 - On lines, joints, and incidences in three dimensions

AU - Elekes, György

AU - Kaplan, Haim

AU - Sharir, Micha

N1 - Funding Information:
✩ Work by Haim Kaplan has been supported by grant 2006/204 from the U.S.–Israel Binational Science Foundation, and by grant 975/06 from the Israel Science Fund. Work by Micha Sharir has been supported by NSF Grants CCF-05-14079 and CCF-08-30272, by grant 2006/194 from the U.S.–Israel Binational Science Foundation, by grants 155/05 and 338/09 from the Israel Science Fund, and by the Hermann Minkowski–MINERVA Center for Geometry at Tel Aviv University. E-mail addresses: haimk@post.tau.ac.il (H. Kaplan), michas@post.tau.ac.il (M. Sharir). 1 Deceased.

PY - 2011/4

Y1 - 2011/4

N2 - We extend (and somewhat simplify) the algebraic proof technique of Guth and Katz (2010) [9], to obtain several sharp bounds on the number of incidences between lines and points in three dimensions. Specifically, we show: (i) The maximum possible number of incidences between n lines in R3 and m of their joints (points incident to at least three non-coplanar lines) is Θ(m1/3n) for mΘn, and Θ(m2/3n2/3+m+n) for mΘn. (ii) In particular, the number of such incidences cannot exceed O(n3/2). (iii) The bound in (i) also holds for incidences between n lines and m arbitrary points (not necessarily joints), provided that no plane contains more than O(n) points and each point is incident to at least three lines. As a preliminary step, we give a simpler proof of (an extension of) the bound O(n3/2), established by Guth and Katz, on the number of joints in a set of n lines in R3. We also present some further extensions of these bounds, and give a trivial proof of Bourgain's conjecture on incidences between points and lines in 3-space, which is an immediate consequence of our incidence bounds, and which constitutes a much simpler alternative to the proof of Guth and Katz (2010) [9].

AB - We extend (and somewhat simplify) the algebraic proof technique of Guth and Katz (2010) [9], to obtain several sharp bounds on the number of incidences between lines and points in three dimensions. Specifically, we show: (i) The maximum possible number of incidences between n lines in R3 and m of their joints (points incident to at least three non-coplanar lines) is Θ(m1/3n) for mΘn, and Θ(m2/3n2/3+m+n) for mΘn. (ii) In particular, the number of such incidences cannot exceed O(n3/2). (iii) The bound in (i) also holds for incidences between n lines and m arbitrary points (not necessarily joints), provided that no plane contains more than O(n) points and each point is incident to at least three lines. As a preliminary step, we give a simpler proof of (an extension of) the bound O(n3/2), established by Guth and Katz, on the number of joints in a set of n lines in R3. We also present some further extensions of these bounds, and give a trivial proof of Bourgain's conjecture on incidences between points and lines in 3-space, which is an immediate consequence of our incidence bounds, and which constitutes a much simpler alternative to the proof of Guth and Katz (2010) [9].

KW - Algebraic techniques

KW - Incidences

KW - Joints

KW - Lines in 3-space

KW - Polynomials

UR - http://www.scopus.com/inward/record.url?scp=78751591447&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2010.11.008

DO - 10.1016/j.jcta.2010.11.008

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AN - SCOPUS:78751591447

SN - 0097-3165

VL - 118

SP - 962

EP - 977

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

IS - 3

ER -