Erd\H{o}s's unit distance problem and rigidity

TL;DR

This paper investigates the maximum number of unit distances among points in the plane, linking the problem to rigidity theory and proposing a structural approach that could improve existing bounds if certain conjectures are proven.

Contribution

It introduces a structural result for nearly extremal point sets and connects the Erdős unit distance problem to a conjecture in rigidity theory, offering a new pathway for progress.

Findings

Established a structural characterization of point sets with near-maximal unit distances.

Reduced the Erdős problem to a conjecture on rigid frameworks.

Identified that proving the conjecture would improve the current bounds.

Abstract

According to a classical result of Spencer, Szemer\'edi, and Trotter (1984), the maximum number of times the unit distance can occur among $n$ points in the plane is $O(n^{4/3})$. This is far from Erd\H{o}s's lower bound, $n^{1+O(1/\log\log n)}$, which is conjectured to be optimal. We prove a structural result for point sets with nearly $n^{4/3}$ unit distances and use it to reduce the problem to a conjecture on rigid frameworks. This conjecture, if true, would yield the first improvement on the bound of Spencer et al. A weaker version of this conjecture has been established by the last two authors.