A note on distinct distances

TL;DR

This paper establishes improved lower bounds on the number of distinct distances determined by points on algebraic curves and varieties, identifying special cases like algebraic helices where bounds differ.

Contribution

It provides new lower bounds for distinct distances on algebraic curves and varieties, refining previous results and introducing a unifying technical lemma.

Findings

Points on algebraic curves span at least  n^{4/3} distinct distances, unless on an algebraic helix.

Points on algebraic varieties have large subsets with  many distinct distances, improving earlier bounds.

A common technical tool underpins both main results, enhancing the theoretical framework.

Abstract

We show that, for a constant-degree algebraic curve $\gamma$ in $\mathbb{R}^D$, every set of $n$ points on $\gamma$ spans at least $\Omega(n^{4/3})$ distinct distances, unless $\gamma$ is an {\it algebraic helix} (see Definition 1.1). This improves the earlier bound $\Omega(n^{5/4})$ of Charalambides [Discrete Comput. Geom. (2014)]. We also show that, for every set $P$ of $n$ points that lie on a $d$-dimensional constant-degree algebraic variety $V$ in $\mathbb{R}^D$, there exists a subset $S\subset P$ of size at least $\Omega(n^{\frac{4}{9+12(d-1)}})$, such that $S$ spans $\binom{|S|}{2}$ distinct distances. This improves the earlier bound of $\Omega(n^{\frac{1}{3d}})$ of Conlon et al. [SIAM J. Discrete Math. (2015)]. Both results are consequences of a common technical tool, given in Lemma 2.7 below.