Distinct distances between a collinear set and an arbitrary set of points

TL;DR

This paper establishes lower bounds on the number of distinct distances between a collinear set of points and an arbitrary set in higher dimensions, under certain geometric restrictions, extending classical distance problems.

Contribution

It provides new lower bounds for the number of distinct distances between a line-based point set and a general point set in any fixed dimension, with geometric constraints.

Findings

Lower bounds depend on sizes of point sets and geometric conditions.

Without restrictions, the number of distinct distances can be linear in total points.

Results extend the understanding of distance problems in higher-dimensional geometry.

Abstract

We consider the number of distinct distances between two finite sets of points in ${\bf R}^k$, for any constant dimension $k\ge 2$, where one set $P_1$ consists of $n$ points on a line $l$, and the other set $P_2$ consists of $m$ arbitrary points, such that no hyperplane orthogonal to $l$ and no hypercylinder having $l$ as its axis contains more than $O(1)$ points of $P_2$. The number of distinct distances between $P_1$ and $P_2$ is then $$ \Omega\left(\min\left\{ n^{2/3}m^{2/3},\; \frac{n^{10/11}m^{4/11}}{\log^{2/11}m},\; n^2,\; m^2\right\}\right) . $$ Without the assumption on $P_2$, there exist sets $P_1$, $P_2$ as above, with only $O(m+n)$ distinct distances between them.