An incomplete attack on the upper bound of the unit distance problem

Steven Senger

An incomplete attack on the upper bound of the unit distance problem

General Mathematics 2026-05-27 v1

Authors: Steven Senger

Abstract

This is an incomplete attempt to show that the upper bound of $\lesssim n^\frac{4}{3}$ on the number unit distances determined by a large finite set of $n$ points in the plane is not sharp. The methods also say something about sets of $n$ points and $n$ lines that attain the sharp bound of the Szemer\'edi-Trotter point-line incidence bound.

Keywords

distance sets incidence geometry of points and lines upper bounds

Cite

@article{arxiv.2605.26145,
  title  = {An incomplete attack on the upper bound of the unit distance problem},
  author = {Steven Senger},
  journal= {arXiv preprint arXiv:2605.26145},
  year   = {2026}
}

Comments

Notes from several years of effort that did not yield a theorem, but might be of independent interest. 6 figures

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