Distinct Distances on Curves via Rigidity

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.

33 pages, to appear in Discrete & Computational Geometry

References in corpus (2)

• On the Erdos distinct distance problem in the plane
• Distinct distances on algebraic curves in the plane

Cited by in corpus (7)

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• Distinct Distances: Open Problems and Current Bounds
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• Improved Elekes-Szabó type estimates using proximity
• A note on distinct distances
• Distinct values of bilinear forms on algebraic curves