An explicit lower bound for the unit distance problem

TL;DR

This paper establishes a new explicit lower bound for the number of unit distance pairs in large point sets in the plane, using advanced number-theoretic constructions.

Contribution

It provides the first explicit construction showing more than n^{1.014} unit distance pairs, improving previous bounds and disproving a longstanding conjecture.

Findings

Sets of n points can have more than n^{1.014} unit distance pairs.

The method uses algebraic number fields with many small primes.

This result improves upon recent non-explicit bounds.

Abstract

We show that there are sets of $n$ points in the plane with $n$ arbitrarily large that contain more than $n^{1.014}$ pairs of points separated by a distance exactly $1$. This improves on very recent work of a team at OpenAI, who proved the same result with an inexplicit exponent greater than $1$, drastically improving on the best previous lower bound and disproving a conjecture of Erd\H{o}s. The method is number-theoretic, relying on constructing algebraic number fields of large degree and small discriminant with many primes of small norm via a Golod-Shafarevich criterion argument.