Distinct distances on non-ruled surfaces and between circles

TL;DR

This paper improves bounds on the number of distinct distances determined by points on non-ruled surfaces in three-dimensional space and characterizes configurations of points on circles with minimal distances, using combinatorial geometry techniques.

Contribution

It provides a new lower bound for distinct distances on non-ruled algebraic surfaces and characterizes when two sets of points on circles in 3D have few distinct distances.

Findings

Points on non-ruled surfaces span .82n^{32/39} distinct distances.

Characterization of circle point sets with O(m+n) distances.

In other cases, the number of distances is at least . m^{2/3}n^{2/3}.

Abstract

We improve the current best bound for distinct distances on non-ruled algebraic surfaces in ${\mathbb R}^3$. In particular, we show that $n$ points on such a surface span $\Omega\left(n^{32/39-\varepsilon}\right)$ distinct distances, for any $\varepsilon>0$. Our proof adapts the proof of Sz\'ekely for the planar case, which is based on the crossing lemma. As part of our proof for distinct distances on surfaces, we also obtain new results for distinct distances between circles in ${\mathbb R}^3$. Consider point sets ${\mathcal P}_1$ and ${\mathcal P}_2$ of respective sizes $m$ and $n$, such that each set lies on a distinct circle in ${\mathbb R}^3$. We characterize the cases when the number of distinct distances between the two sets can be $O(m+n)$. This includes a new configuration with a small number of distances. In any other case, we prove that the number of distinct distances is $\Omega\left(\min\left\{m^{2/3}n^{2/3},m^2,n^2\right\}\right)$.