Abstract
We discuss a unified approach to a class of geometric combinatorics incidence problems in $2D$, of the Erdös distance type. The goal is obtaining the second moment estimate, that is given a finite point set $S$ and a function $f$ on $S\times S$, an upper bound on the number of solutions of $$ f(p,p') = f(q,q')\neq 0,\qquad (p,p',q,q')\in S\times S\times S\times S. \qquad(*) $$ E.g., $f$ is the Euclidean distance in the plane, sphere, or a sheet of the twosheeted hyperboloid. Our tool is the GuthKatz incidence theorem for lines in $\mathbb{RP}^3$, but we focus on how the original $2D$ problem is made amenable to it. This procedure was initiated by Elekes and Sharir, based on symmetry considerations. However, symmetry considerations can be bypassed or made implicit. The classical PlückerKlein formalism for line geometry enables one to directly interpret a solution of $(*)$ as intersection of two lines in $\mathbb{RP}^3$. This allows for a very brief argument extending the Euclidean plane distance argument to the spherical and hyperbolic distances. We also find instances of the question $(*)$ without underlying symmetry group. The space of lines in the threespace, the Klein quadric $\mathcal K$, is fourdimensional. We start out with an injective map $\mathfrak F:\,S\times S\to\mathcal K$, from a pair of points in $2D$ to a line in $3D$ and seek a combinatorial problem in the form $(*)$, which can be solved by applying the GuthKatz theorem to the set of lines in question. We identify a few new such problems and generalise the existing ones.
Original language  English 

Pages (fromto)  934954 
Number of pages  21 
Journal  SIAM Journal on Discrete Mathematics 
Volume  30 
Issue number  2 
Early online date  12 May 2016 
DOIs  
Publication status  Published  Jun 2016 
Bibliographical note
Theorem 5', implicit in the earlier verisons has been stated explicitly in this ArXiv version, giving a family of applications of the GuthKatz theorem to sumproduct type quantities, with no underlying symmetry groupKeywords
 math.MG
 math.CO
 68R05, 11B75
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Dr Misha Rudnev
 School of Mathematics  Associate Professor in Mathematics
 Number theory and combinatorics
 Pure Mathematics
Person: Academic , Member