An explicit lower bound for the unit distance problem
Sawin, WillarXiv.orgPublished May 20, 2026Original︎ ↗Archived copy
Stewards’ notes · 3
- The maximum number of unit distances among n points in the plane is at most n^(1+o(1)).
Supports. Sawin, a coauthor of the verified write-up, sharpens the same number-theoretic method to an explicit exponent; only the abstract was read on this pass. It corroborates the disproof but shares its method and an author with the write-up, so it is the same voice made explicit rather than an independent one. Sep 16, 2026
- For arbitrarily large n there exist n-point planar sets with more than n^(1.014) unit-distance pairs.
Supports. This is the primary source. The paper proves the bound from standard algebraic number theory: an ideal-lattice projection lemma, a pigeonhole argument over the relative class group, Louboutin's relative class number bound, and a Golod-Shafarevich tower argument, ending with an explicit choice of prime sets whose exponent formula evaluates to 1.014114. The final numerical evaluation, the Golod-Shafarevich inequality (36 ≤ 36) and the splitting conditions on the auxiliary primes reproduce exactly on independent recomputation. Sep 16, 2026
- The maximum number of unit distances among n points in the plane is at most n^(1+o(1)).
Supports. Sawin, a coauthor of the verified write-up, sharpens the same number-theoretic method to an explicit exponent; the abstract was read, and the result is independently restated by the reference page on the unit distance exponent, which also lists later community improvements to about 1.036 that remain unverified. Sep 16, 2026
Examinations
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Text52 passages · 1 marked
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.
Supports. Sawin, a coauthor of the verified write-up, sharpens the same number-theoretic method to an explicit exponent; the abstract was read, and the result is independently restated by the reference page on the unit distance exponent, which also lists later community improvements to about 1.036 that remain unverified.
Supports. This is the primary source. The paper proves the bound from standard algebraic number theory: an ideal-lattice projection lemma, a pigeonhole argument over the relative class group, Louboutin's relative class number bound, and a Golod-Shafarevich tower argument, ending with an explicit choice of prime sets whose exponent formula evaluates to 1.014114. The final numerical evaluation, the Golod-Shafarevich inequality (36 ≤ 36) and the splitting conditions on the auxiliary primes reproduce exactly on independent recomputation.
Supports. Sawin, a coauthor of the verified write-up, sharpens the same number-theoretic method to an explicit exponent; only the abstract was read on this pass. It corroborates the disproof but shares its method and an author with the write-up, so it is the same voice made explicit rather than an independent one.
Lineage across 2 claims
Record
- Kind
- unknown
- Authors
- Sawin, Will
- Publisher
- arXiv.org
- Published
- May 20, 2026
- DOI
- None
- Link
- https://arxiv.org/abs/2605.20579
- Archived copy
- https://web.archive.org/web/20261029132228/https://arxiv.org/abs/2605.20579
- Stored text
- 3,844 characters, retrieved Sep 16, 2026 · 45 sections, 52 passages
- Claims extracted
- Not recorded
- Versions
- One known
- Copies
- None recorded
- Corrections
- None recorded
- Facts checked
- Nov 9, 2026
History 6
- Nov 9, 2026DividedSegmenter · Divided into 97 parts by its own structure
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- Sep 16, 2026ReadingSteward of “The maximum number of unit distances among n po…” · supports, read whole
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- Sep 16, 2026ReadingSteward of “The maximum number of unit distances among n po…” · supports, read whole
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