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\[D(\mathcal{P}_1,\mathcal{P}_2) = \begin{cases}\Omega(m^{1/2}n^{1/2}\log^{-1/2}n), \ \ & \mbox{ when } m = \Omega(n^{1/2}\log^{-1/3}n), \\\Omega(m^{1/3}n^{1/2}), \ \ & \mbox{ when } m=O(n^{1/2}\log^{-1/3}n). \end{cases}\]This generalizes work of Pohoata and Sheffer, and complements work of Pach and de Zeeuw.
@article{10_37236_8956,
author = {Bryce McLaughlin and Mohamed Omar},
title = {On distinct distances between a variety and a point set},
journal = {The electronic journal of combinatorics},
year = {2022},
volume = {29},
number = {3},
doi = {10.37236/8956},
zbl = {1494.52016},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8956/}
}
TY - JOUR
AU - Bryce McLaughlin
AU - Mohamed Omar
TI - On distinct distances between a variety and a point set
JO - The electronic journal of combinatorics
PY - 2022
VL - 29
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/8956/
DO - 10.37236/8956
ID - 10_37236_8956
ER -
%0 Journal Article
%A Bryce McLaughlin
%A Mohamed Omar
%T On distinct distances between a variety and a point set
%J The electronic journal of combinatorics
%D 2022
%V 29
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/8956/
%R 10.37236/8956
%F 10_37236_8956
Bryce McLaughlin; Mohamed Omar. On distinct distances between a variety and a point set. The electronic journal of combinatorics, Tome 29 (2022) no. 3. doi: 10.37236/8956
