Distinct Distances for Points Lying on Curves in \({\mathbb R}^\boldsymbol{d}\)—The Bipartite Case
Hadas Baer-Erenfeld, Orit E. Raz · SIAM Journal on Discrete Mathematics · 2025
Abstract. Let [Formula: see text] be a pair of constant-degree irreducible algebraic curves in [Formula: see text]. Assume that [Formula: see text] is contained in neither a hyperplane nor a quadric surface in [Formula: see text] for each [Formula: see text]. We show that for every pair of [Formula: see text]-point sets [Formula: see text] and [Formula: see text], the number of distinct distances spanned by [Formula: see text] is [Formula: see text] with a constant of proportionality that depends on [Formula: see text], [Formula: see text], and [Formula: see text]. 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 [Formula: see text] of the complete bipartite graph [Formula: see text] that are not infinitesimally rigid.–