DISTINCT DISTANCES FOR POINTS LYING ON CURVES IN R^d---THE BIPARTITE CASE

Let γ₁, γ₂ be a pair of constant-degree irreducible algebraic curves in R^d . Assume that γ_i is contained in neither a hyperplane nor 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 x P₂ is Ω(n^3/2) with a constant of proportionality that depends on deg γ₁, deg γ₂, and d. This extends earlier results of Charalambides [Discrete Comput. Geom., 51 (2014), pp. 666--701], Pach and de Zeeuw [Combin. Probab. Comput., 26 (2017), pp. 99--117], and Raz [Combin. Probab. Comput., 29 (2020), pp. 650--663] to the bipartite version. For the proof we use rigidity theory, and in particular the description of Bolker and Roth [Pacific J. Math., 90 (1980), pp. 27--44] for realizations in R^d of the complete bipartite graph K_m,n that are not infinitesimally rigid.--.