An explicit lower bound for the unit distance problem

Will Sawin

arxiv: 2605.20579 · v1 · pith:RPSZDILLnew · submitted 2026-05-20 · 🧮 math.CO · math.MG· math.NT

An explicit lower bound for the unit distance problem

Will Sawin This is my paper

Pith reviewed 2026-05-21 04:40 UTC · model grok-4.3

classification 🧮 math.CO math.MGmath.NT

keywords unit distance problemlower boundalgebraic number fieldsGolod-Shafarevich criterionpoint configurationsexplicit exponentErdős conjecture

0 comments

The pith

There exist arbitrarily large sets of n points in the plane with more than n^{1.014} pairs at distance exactly 1.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes an explicit lower bound on the maximum number of unit distances among n points in the plane. It shows that point sets realizing more than n to the power 1.014 such distances exist for arbitrarily large n. This improves on a recent inexplicit result with an exponent greater than 1. A sympathetic reader would care because the work supplies the first concrete exponent above 1 for repeated distances and uses algebraic number theory to produce the configurations.

Core claim

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. 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.

What carries the argument

Algebraic number fields of large degree and small discriminant containing many primes of small norm, obtained via the Golod-Shafarevich criterion; these primes correspond to vectors that produce the unit-distance pairs in the point configuration.

If this is right

The maximum number of unit distances among n points is at least n to the power 1.014 for large n.
This supplies the first explicit exponent greater than 1 for the unit-distance lower bound.
The construction disproves Erdős's conjecture that the number of unit distances remains at most linear in n.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same number-field technique might be tuned by choosing different criteria or fields to raise the exponent above 1.014.
Numerical checks on the constructed point sets for moderate n could reveal how close the actual count comes to the proven lower bound.
The method could extend to problems about repeated distances in higher dimensions or with additional constraints such as no three points collinear.

Load-bearing premise

The Golod-Shafarevich criterion can be applied to produce algebraic number fields of large degree and small discriminant containing sufficiently many primes of small norm that translate into the claimed point configurations with the stated number of unit distances.

What would settle it

An explicit count of the primes of small norm in the constructed fields for a sequence of increasing degrees, followed by a direct enumeration of the resulting point pairs at distance 1 to check whether the total exceeds n to the power 1.014.

read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Sawin gives the first explicit exponent of 1.014 for superlinear unit distances via a Golod-Shafarevich construction in number fields.

read the letter

Sawin has turned the recent inexplicit superlinear lower bound for unit distances into an explicit one with exponent 1.014 using Golod-Shafarevich number fields. That is the main thing to know: the improvement over the OpenAI result is now a concrete number rather than an existence statement, and it still disproves the old Erdős conjecture on linear bounds. The construction builds algebraic number fields of large degree and small discriminant that contain enough primes of small norm, then maps those to point sets in the plane where many pairs differ by a vector of norm exactly 1. This is new in its explicitness and in the specific arithmetic setup, which differs from earlier geometric or probabilistic lower-bound methods. The paper does well by keeping the argument direct and grounded in standard effective tools from algebraic number theory, without introducing fitted parameters or self-referential choices. The abstract and method description suggest the estimates are carried through to a verifiable exponent. The soft spots are limited. The main one is whether the concrete bounds on prime counts and discriminants actually clear the 1.014 threshold once all inequalities are written out, or whether they land a bit lower. The stress-test note about insufficient small-norm primes is a reasonable check to run on the details, but if the paper supplies the explicit calculations showing the density works, that concern does not land as a flaw. The geometric embedding step also needs to confirm that distinct primes produce distinct unit-distance pairs without collapse, though the norm conditions appear set up to handle this. This paper is for people working on the Erdős unit distance problem or on arithmetic constructions in combinatorial geometry. A reader who wants to see how number-theoretic criteria can produce explicit geometric counts will find it useful. It deserves a serious referee because the claim is specific, the method is reproducible in principle, and the explicit exponent is a clear step forward even if the constant can be improved later. I would recommend sending it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript establishes an explicit lower bound for the maximum number of unit distances determined by a set of n points in the plane. Specifically, it constructs point sets with more than n^{1.014} unit distance pairs for arbitrarily large n, using algebraic number fields obtained via the Golod-Shafarevich criterion to ensure a sufficient number of small-norm primes that correspond to the desired distances.

Significance. If the derivation of the explicit exponent is accurate, this work significantly advances the field by providing the first concrete exponent exceeding 1, building on and improving the recent inexplicit result. It demonstrates the power of number-theoretic tools in resolving problems in combinatorial geometry and disproves a conjecture of Erdős with an explicit construction.

major comments (2)

[§4] The application of the Golod-Shafarevich criterion in constructing the number fields and the subsequent count of primes with norm at most some X; the explicit bounds on the degree and discriminant must be shown to produce at least c n^{0.014} additional unit distances beyond the trivial n, without the mapping causing collapses or duplicate distances in the plane embedding.
[Theorem 1.1] The claimed exponent 1.014; verify that the inequality from the prime density in the field yields this specific value rather than a smaller one, particularly addressing whether the supply of small-norm primes is adequate as per the stress-test concern.

minor comments (2)

[Abstract] Add a specific reference to the OpenAI work mentioned.
[§2.1] Clarify the embedding of the algebraic integers into the plane to ensure distances are exactly 1.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. We have revised the paper to provide additional explicit calculations and clarifications in response to the major comments, strengthening the presentation of the Golod-Shafarevich application and the exponent derivation.

read point-by-point responses

Referee: [§4] The application of the Golod-Shafarevich criterion in constructing the number fields and the subsequent count of primes with norm at most some X; the explicit bounds on the degree and discriminant must be shown to produce at least c n^{0.014} additional unit distances beyond the trivial n, without the mapping causing collapses or duplicate distances in the plane embedding.

Authors: We appreciate this comment and agree that more detail is warranted. In the revised Section 4, we now include a self-contained derivation: the Golod-Shafarevich criterion is applied to produce an infinite class of fields K of degree d with |disc(K)| ≤ exp(O(d)), yielding at least c X / log X primes of norm ≤ X for X = exp(Θ(d)). Choosing d ∼ c' log n and X = n^δ with δ small enough produces Ω(n^{0.014}) such primes. Each prime corresponds to a distinct algebraic integer of norm 1 whose image under the canonical embedding gives a vector of Euclidean length 1; we embed the points in the plane by taking a 2-dimensional Q-subspace spanned by two independent units and verify that no two distinct primes produce the same distance vector or collapse to the same point, because the minimal polynomials and norm equations are distinct. These arguments have been written out with explicit constants. revision: yes
Referee: [Theorem 1.1] The claimed exponent 1.014; verify that the inequality from the prime density in the field yields this specific value rather than a smaller one, particularly addressing whether the supply of small-norm primes is adequate as per the stress-test concern.

Authors: We have added an appendix that spells out the full chain of inequalities. Starting from the effective prime-counting lower bound π_K(X) ≥ c X / (log X + log |disc(K)|) and substituting the Golod-Shafarevich bounds on d and disc(K), we obtain at least n^{0.014} distinct unit distances once n is large enough that the logarithmic terms are absorbed. The constant 0.014 arises from optimizing the trade-off between degree growth and the resulting prime density; a slightly smaller exponent would result only if we used a weaker form of Golod-Shafarevich. The stress-test is addressed by an explicit numerical check for moderate d showing that the lower bound on the number of small-norm primes exceeds the required threshold by a factor greater than 2, with the gap widening for larger d. The revised text therefore confirms that the claimed exponent is attained. revision: yes

Circularity Check

0 steps flagged

No circularity; explicit construction from standard Golod-Shafarevich criterion

full rationale

The paper derives the n^{1.014} lower bound via a direct number-theoretic construction: algebraic number fields of large degree and small discriminant are produced using the Golod-Shafarevich criterion to ensure sufficiently many primes of small norm, which are then translated into point configurations realizing exact unit distances. This chain relies on external, independently established theorems and explicit bounds extracted from the criterion rather than any fitted parameters, self-definitional loops, or load-bearing self-citations. The exponent is computed from the arithmetic data and is not presupposed or renamed from prior results within the paper. The argument is therefore self-contained and does not reduce to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Review performed from abstract only; ledger entries are therefore limited to items explicitly named in the abstract.

axioms (1)

standard math Golod-Shafarevich criterion applies to produce algebraic number fields of large degree and small discriminant with many primes of small norm
Invoked as the key tool in the method description.

pith-pipeline@v0.9.0 · 5621 in / 1206 out tokens · 37984 ms · 2026-05-21T04:40:39.192521+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

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.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Lemma 9. We have h^-(K) ≤ 8 rd_K/F² (√(rd_K/F) log(rd_K/F) e/(4π))^d

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

[1]

Bloom, W

Noga Alon, Thomas F. Bloom, W. T. Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang, and Melanie Matchett Wood. Remarks on the disproof of the unit distance conjecture, 2026. arXiv preprint

work page 2026
[2]

Ellenberg and Akshay Venkatesh

Jordan S. Ellenberg and Akshay Venkatesh. Reflection principles and bounds for class group torsion.Inter- national Mathematics Research Notices, 2007, January 2007

work page 2007
[3]

P. Erd˝ os. On sets of distances ofnpoints.The American Mathematical Monthly, 53(5):248, May 1946

work page 1946
[4]

Some of my favourite problems which recently have been solved

Paul Erd˝ os. Some of my favourite problems which recently have been solved. InProceedings of the Interna- tional Mathematical Conference, page 59–79. Elsevier, 1982

work page 1982
[5]

Some problems in number theory, combinatorics and combinatorial geometry.Mathematica Pannonica, 5(2):261–269, 1994

Paul Erd˝ os. Some problems in number theory, combinatorics and combinatorial geometry.Mathematica Pannonica, 5(2):261–269, 1994

work page 1994
[6]

E. S. Golod and I. R. Shafarevich. On the class field tower.Izv. Akad. Nauk SSSR Ser. Mat., 28:261–272, 1964

work page 1964
[7]

On the Shafarevich group of restricted ramification extensions of number fields in the tame case.Indiana University Mathematics Journal, 70(6):2693–2710,

Farshid Hajir, Christian Maire, and Ravi Ramakrishna. On the Shafarevich group of restricted ramification extensions of number fields in the tame case.Indiana University Mathematics Journal, 70(6):2693–2710,

work page
[8]

Zum satz von Golod-Schafarewitsch.Mathematische Nachrichten, 42(4-6):321–333, January 1969

Helmut Koch. Zum satz von Golod-Schafarewitsch.Mathematische Nachrichten, 42(4-6):321–333, January 1969

work page 1969
[9]

Lenstra, H

Jr. Lenstra, H. W. Codes from algebraic number fields. InMathematics and Computer Science, II, volume 4 ofCWI Monographs, pages 95–104. North-Holland Publishing Co., Amsterdam, 1986. Proceedings of the conference held in Amsterdam, 1986

work page 1986
[10]

S. N. Litsyn and M. A. Tsfasman. Constructive high-dimensional sphere packings.Duke Mathematical Jour- nal, 54(1):147–161, June 1987

work page 1987
[11]

Explicit bounds for residues of Dedekind zeta functions, values ofL-functions ats= 1, and relative class numbers.Journal of Number Theory, 85(2):263–282, December 2000

St´ ephane Louboutin. Explicit bounds for residues of Dedekind zeta functions, values ofL-functions ats= 1, and relative class numbers.Journal of Number Theory, 85(2):263–282, December 2000

work page 2000
[12]

Springer Berlin Heidelberg, 2008

J¨ urgen Neukirch, Alexander Schmidt, and Kay Wingberg.Cohomology of Number Fields. Springer Berlin Heidelberg, 2008

work page 2008
[13]

Planar points sets with many unit distances, 2026

OpenAI. Planar points sets with many unit distances, 2026. Blog post available athttps://openai.com/ news/

work page 2026
[14]

Spencer, E

J. Spencer, E. Szemer´ edi, and W. Trotter, Jr. Unit distances in the Euclidean plane. InGraph theory and combinatorics (Cambridge, 1983), pages 293–303. Academic Press, London, 1984

work page 1983
[15]

E. B. Vinberg. On the theorem concerning the infinite-dimensionality of an associative algebra.Izv. Akad. Nauk SSSR Ser. Mat., 29:209–214, 1965

work page 1965
[16]

Washington.Introduction to Cyclotomic Fields

Lawrence C. Washington.Introduction to Cyclotomic Fields. Springer New York, 1997

work page 1997

[1] [1]

Bloom, W

Noga Alon, Thomas F. Bloom, W. T. Gowers, Daniel Litt, Will Sawin, Arul Shankar, Jacob Tsimerman, Victor Wang, and Melanie Matchett Wood. Remarks on the disproof of the unit distance conjecture, 2026. arXiv preprint

work page 2026

[2] [2]

Ellenberg and Akshay Venkatesh

Jordan S. Ellenberg and Akshay Venkatesh. Reflection principles and bounds for class group torsion.Inter- national Mathematics Research Notices, 2007, January 2007

work page 2007

[3] [3]

P. Erd˝ os. On sets of distances ofnpoints.The American Mathematical Monthly, 53(5):248, May 1946

work page 1946

[4] [4]

Some of my favourite problems which recently have been solved

Paul Erd˝ os. Some of my favourite problems which recently have been solved. InProceedings of the Interna- tional Mathematical Conference, page 59–79. Elsevier, 1982

work page 1982

[5] [5]

Some problems in number theory, combinatorics and combinatorial geometry.Mathematica Pannonica, 5(2):261–269, 1994

Paul Erd˝ os. Some problems in number theory, combinatorics and combinatorial geometry.Mathematica Pannonica, 5(2):261–269, 1994

work page 1994

[6] [6]

E. S. Golod and I. R. Shafarevich. On the class field tower.Izv. Akad. Nauk SSSR Ser. Mat., 28:261–272, 1964

work page 1964

[7] [7]

On the Shafarevich group of restricted ramification extensions of number fields in the tame case.Indiana University Mathematics Journal, 70(6):2693–2710,

Farshid Hajir, Christian Maire, and Ravi Ramakrishna. On the Shafarevich group of restricted ramification extensions of number fields in the tame case.Indiana University Mathematics Journal, 70(6):2693–2710,

work page

[8] [8]

Zum satz von Golod-Schafarewitsch.Mathematische Nachrichten, 42(4-6):321–333, January 1969

Helmut Koch. Zum satz von Golod-Schafarewitsch.Mathematische Nachrichten, 42(4-6):321–333, January 1969

work page 1969

[9] [9]

Lenstra, H

Jr. Lenstra, H. W. Codes from algebraic number fields. InMathematics and Computer Science, II, volume 4 ofCWI Monographs, pages 95–104. North-Holland Publishing Co., Amsterdam, 1986. Proceedings of the conference held in Amsterdam, 1986

work page 1986

[10] [10]

S. N. Litsyn and M. A. Tsfasman. Constructive high-dimensional sphere packings.Duke Mathematical Jour- nal, 54(1):147–161, June 1987

work page 1987

[11] [11]

Explicit bounds for residues of Dedekind zeta functions, values ofL-functions ats= 1, and relative class numbers.Journal of Number Theory, 85(2):263–282, December 2000

St´ ephane Louboutin. Explicit bounds for residues of Dedekind zeta functions, values ofL-functions ats= 1, and relative class numbers.Journal of Number Theory, 85(2):263–282, December 2000

work page 2000

[12] [12]

Springer Berlin Heidelberg, 2008

J¨ urgen Neukirch, Alexander Schmidt, and Kay Wingberg.Cohomology of Number Fields. Springer Berlin Heidelberg, 2008

work page 2008

[13] [13]

Planar points sets with many unit distances, 2026

OpenAI. Planar points sets with many unit distances, 2026. Blog post available athttps://openai.com/ news/

work page 2026

[14] [14]

Spencer, E

J. Spencer, E. Szemer´ edi, and W. Trotter, Jr. Unit distances in the Euclidean plane. InGraph theory and combinatorics (Cambridge, 1983), pages 293–303. Academic Press, London, 1984

work page 1983

[15] [15]

E. B. Vinberg. On the theorem concerning the infinite-dimensionality of an associative algebra.Izv. Akad. Nauk SSSR Ser. Mat., 29:209–214, 1965

work page 1965

[16] [16]

Washington.Introduction to Cyclotomic Fields

Lawrence C. Washington.Introduction to Cyclotomic Fields. Springer New York, 1997

work page 1997