# On Distinct Distances Between a Variety and a Point Set

## TL;DR

This paper establishes new lower bounds on the number of distinct distances between a point set on a fixed-degree algebraic curve and an arbitrary point set in the plane, generalizing previous results.

## Contribution

It provides novel lower bounds on distinct distances involving a curve-constrained point set and an arbitrary set, extending prior work in combinatorial geometry.

## Key findings

- Lower bound: (\u211c_1,_2) = (m^{1/2}n^{1/2}\u2212  n^{1/2}^{-1/2})
- Lower bound: (_1,_2) = (n^{1/2} m^{1/3})
- Results generalize previous bounds by Pohoata, Sheffer, Pach, and de Zeeuw.

## Abstract

We consider the problem of determining the number of distinct distances between two point sets in $\mathbb{R}^2$ where one point set $\mathcal{P}_1$ of size $m$ lies on a real algebraic curve of fixed degree $r$, and the other point set $\mathcal{P}_2$ of size $n$ is arbitrary. We prove that the number of distinct distances between the point sets, $D(\mathcal{P}_1,\mathcal{P}_2)$, satisfies $D(\mathcal{P}_1,\mathcal{P}_2) = \Omega(m^{1/2}n^{1/2}\log^{-1/2}n)$ when $m = \Omega(n^{1/2}\log^{-1/3}n)$ and $D(\mathcal{P}_1,\mathcal{P}_2) = \Omega(n^{1/2} m^{1/3})$ when $m=O(n^{1/2}\log^{-1/3}n)$ This generalizes work of Pohoata and Sheffer, and complements work of Pach and de Zeeuw.