[2605.20579] An explicit lower bound for the unit distance problem

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Mathematics > Combinatorics

arXiv:2605.20579 (math)

[Submitted on 20 May 2026]

Title:An explicit lower bound for the unit distance problem

Authors:Will Sawin<br>View a PDF of the paper titled An explicit lower bound for the unit distance problem, by Will Sawin

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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 Erdős. 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.

Subjects:

Combinatorics (math.CO); Metric Geometry (math.MG); Number Theory (math.NT)

Cite as:<br>arXiv:2605.20579 [math.CO]

(or<br>arXiv:2605.20579v1 [math.CO] for this version)

https://doi.org/10.48550/arXiv.2605.20579

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arXiv-issued DOI via DataCite (pending registration)

Submission history<br>From: Will Sawin [view email]<br>[v1]<br>Wed, 20 May 2026 00:37:35 UTC (17 KB)

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