On Bipartite Distinct Distances in the Plane

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 showed that $D(m, n) = O(\sqrt{mn})$ when $m \leqslant n^{1/3}$. For $m \geqslant n^{1/3}$, 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 $m \leqslant n^{1/3}$. This is done by adapting Székely'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 $m \geqslant n^{1/3}$.

On Distinct Distances Between a Variety and a Point Set

“…Theorem 1 matches Elekes' upper bound when m = Ω(n 1/2 log −1/3 n), up to a factor of log 1/2 n. This work is benefited by recent results of Pohoata and Sheffer [11] that establishes similar lower bounds for D(P 1 , P 2 ) when P 1 is restricted to a line and P 2 is arbitrary. We note here that since the submission of this article, Mathialagan [8] has developed a further generalization that supersedes the work of Pohoata and Sheffer and the work here, providing a bound for D(P 1 , P 2 ) when both P 1 and P 2 are unrestricted.…”