Distinct distances for points lying on curves in $\mathbb{R}^d$ -- the bipartite case

Abstract

Let $\gamma_1,\gamma_2$ be a pair of constant-degree irreducible algebraic curves in $\mathbb{R}^d$. Assume that $\gamma_i$ is neither contained in a hyperplane nor in a quadric surface in $\mathbb{R}^d$, for each $i=1,2$. We show that for every pair of $n$-point sets $P_1\subset\gamma_1$ and $P_2\subset\gamma_2$, the number of distinct distances spanned by $P_1\times P_2$ is $\Omega(n^{3/2})$, with a constant of proportionality that depends on ${\rm deg}\gamma_1$, ${\rm deg}\gamma_2$, 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 $\mathbb{R}^d$ of the complete bipartite graph $K_{m,n}$ that are not infinitesimally rigid.