Distinct distances for points lying on curves in ℝ^d -- the bipartite case

2023/04/13 by Baer-Erenfeld, Hadas, Raz, Orit E.
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2304.06812

Abstract

Let γ₁,γ₂ be a pair of constant-degree irreducible algebraic curves in ℝ^d. Assume that γ_i is neither contained in a hyperplane nor in a quadric surface in ℝ^d, for each i=1,2. We show that for every pair of n-point sets P₁⊂γ₁ and P₂⊂γ₂, the number of distinct distances spanned by P₁× P₂ is Ω(n^3/2), with a constant of proportionality that depends on \rm degγ₁, \rm degγ₂, and d. This extends earlier results of Charalambides [Char], Pach and De Zeeuw [PdZ], and Raz [Ra] to the bipartite version. For the proof we use rigidity theory, and in particular the description of Bolker and Roth [BR80] for realizations in ℝ^d of the complete bipartite graph K_m,n that are not infinitesimally rigid.

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