On Bipartite Distinct Distances in the Plane

TL;DR

This paper establishes tight bounds on the minimum number of distinct bipartite distances in the plane for sets of points of different sizes, extending classical results and employing combinatorial geometry techniques.

Contribution

It proves the lower bounds matching previous upper bounds for bipartite distances, adapting Székely's crossing number argument and extending Guth and Katz's analysis.

Findings

Lower bound $D(m,n) = oldsymbol{ ext{Ω}}(\sqrt{mn})$ for $m oldsymbol{ ext{≤}} n^{1/3}$

Lower bound $D(m,n) = oldsymbol{ ext{Ω}}(rac{\sqrt{mn}}{ ext{log} ext{n}})$ for $m oldsymbol{ ext{≥}} n^{1/3}$

Elekes' construction is tight for the bipartite distance problem.

Abstract

Given sets $\mathcal{P}, \mathcal{Q} \subseteq \mathbb{R}^2$ of sizes $m$ and $n$ respectively, we are interested in the number of distinct distances spanned by $\mathcal{P} \times \mathcal{Q}$. Let $D(m, n)$ denote the minimum number of distances determined by sets in $\mathbb{R}^2$ of sizes $m$ and $n$ respectively, where $m \leq n$. Elekes \cite{CircleGrids} showed that $D(m, n) = O(\sqrt{mn})$ when $m \leq n^{1/3}$. For $m \geq n^{1/3}$, we have the upper bound $D(m, n) = O(n/\sqrt{\log n})$ as in the classical distinct distances problem. In this work, we show that Elekes' construction is tight by deriving the lower bound of $D(m, n) = \Omega(\sqrt{mn})$ when $m \leq n^{1/3}$. This is done by adapting Sz\'{e}kely's crossing number argument. We also extend the Guth and Katz analysis for the classical distinct distances problem to show a lower bound of $D(m, n) = \Omega(\sqrt{mn}/\log n)$ when $m \geq n^{1/3}$.