An improved constant factor for the unit distance problem

TL;DR

This paper improves the upper bound on the maximum number of unit distances among n points in the plane to approximately 1.94n^{4/3}, refining previous bounds and exploring related geometric bounds and crossing lemma variants.

Contribution

The paper presents a tighter upper bound for the unit distance problem and introduces improved constants for related geometric inequalities.

Findings

Upper bound for unit distances: 1.94·n^{4/3}

Enhanced bounds for small n values

Refined crossing lemma constants

Abstract

We prove that the number of unit distances among $n$ planar points is at most $1.94\cdot n^{4/3}$, improving on the previous best bound of $8n^{4/3}$. We also give better upper and lower bounds for several small values of $n$. We also prove some variants of the crossing lemma and improve some constant factors.