Distinct Distances on Curves via Rigidity

TL;DR

The paper proves that points on most algebraic curves in real space determine significantly more distinct distances than on special curves like lines or tori, using rigidity theory to establish these bounds.

Contribution

It introduces a novel approach linking rigidity of frameworks on curves to lower bounds on the number of distinct distances.

Findings

Points on algebraic curves generally determine  N^{1+1/4} distinct distances.

Special curves like lines or tori allow only  N distances.

The method applies to other geometric quantities to find similar bounds.

Abstract

It is shown that $N$ points on a real algebraic curve of degree $n$ in $\mathbb{R}^d$ always determine $\gtrsim_{n,d}N^{1+\frac{1}{4}}$ distinct distances, unless the curve is a straight line or the closed geodesic of a flat torus. In the latter case, there are arrangements of $N$ points which determine $\lesssim N$ distinct distances. The method may be applied to other quantities of interest to obtain analogous exponent gaps. An important step in the proof involves understanding the structural rigidity of certain frameworks on curves.