Distinct distances on two lines

2013/02/13 by Micha Sharir, Sharir, Micha, Adam Sheffer +3 · 1 citation
Computer Science · Mathematics · #Combinatorics (math.CO) #Computational Geometry (cs.CG) #FOS: Computer and information sciences #FOS: Mathematics #cs.CG #math.CO

paper · pdf · doi:10.48550/arxiv.1302.3081

arxiv created 2013/06/02 · arxiv updated 2013/06/04

Abstract

Let P₁ and P₂ be two sets of points in the plane, so that P₁ is contained in a line L₁, P₂ is contained in a line L₂, and L₁ and L₂ are neither parallel nor orthogonal. Then the number of distinct distances determined by the pairs of P₁xP₂ is Ω(min|P₁|^2/3|P₂|^2/3,|P₁|², |P₂|²). In particular, if |P₁|=|P₂|=m, then the number of these distinct distances is Ω(m^4/3), improving upon the previous bound Ω(m^5/4) of Elekes.

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