Pith. sign in

REVIEW 2 major objections 2 minor 27 cited by

Remarks on the disproof of the unit distance conjecture

T0 review · 2 major / 2 minor · reviewed 2026-05-21 · grok-4.3

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read A human-verified algebraic point set disproves the Erdős unit distance conjecture.

desk verdict This is a careful human digest and reflection on an externally generated counterexample to the Erdős unit distance conjecture rather than an independent construction or proof. read the letter →

arxiv 2605.20695 v1 pith:HTMKJTNB submitted 2026-05-20 math.CO math.NT

Noga Alon , Thomas F. Bloom , W. T. Gowers , Daniel Litt , Will Sawin , Arul Shankar , Jacob Tsimerman , Victor Wang
show 1 more author
Melanie Matchett Wood
This is my paper · ORCID
classification math.COmath.NT
keywords Erdősunitdistanceconjecturecounterexampledistancespointconfigurationsalgebraicconstructionhumanverification
checked against alexander_duality_circle_linkingFoundation.DimensionForcing
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper supplies a compact, hand-checkable version of a counterexample to the Erdős unit distance conjecture, which had claimed that n points in the plane determine at most O(n to the 4/3) unit distances. The authors condense an earlier AI-generated construction into a short argument that draws on algebraic and number-theoretic techniques traceable to Ellenberg-Venkatesh, the Golod-Shafarevich theorem, and Hajir-Maire-Ramakrishna. A reader following the argument sees an explicit finite collection of points whose pairwise distances include more than the conjectured number of exact units. The work therefore shows that the long-standing upper bound does not hold.

What carries the argument

An algebraic construction of a point configuration, obtained by solving systems of polynomial equations or group presentations that force many pairs of points to satisfy the unit-distance equation.

What would settle it

An independent enumeration of all pairwise distances in the presented point set that shows the total number of exact unit distances is strictly larger than the maximum allowed by the Erdős conjecture for that number of points.

Watch

Extended reading notes

Core claim

The authors exhibit a finite point set in the Euclidean plane that realizes more unit distances than the Erdős conjecture permits, presenting the construction in a form short enough for direct human verification and built from algebraic ideas associated with Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna.

Load-bearing premise

The original AI-generated construction contains no undetected algebraic or geometric errors and the human verification step has correctly established that the resulting point set exceeds the conjectured number of unit distances.

Editorial extensions

If this is right

  • The maximum number of unit distances among n points in the plane must grow faster than the conjectured O(n^{4/3}) bound.
  • New finite point configurations exist that realize asymptotically more unit distances than previously thought possible.
  • The same style of algebraic construction may be adapted to produce high-distance examples in other metric spaces or for other fixed distances.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • AI systems can generate candidate constructions that become transparent once reduced to their essential algebraic steps.
  • The same circle of ideas might be tested on related open questions about incidences or repeated distances in the plane.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a short, digested, human-verified version of an OpenAI-generated counterexample to the Erdős unit distance conjecture. It claims the existence of a finite point set in the plane realizing strictly more than O(n^{4/3}) unit distances and traces key steps of the argument to ideas from Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna, followed by reflections on the construction.

Significance. A verified counterexample would disprove a central conjecture in discrete geometry and combinatorial number theory. The paper's contribution is in distilling the AI-generated example into a human-accessible form and linking it to prior algebraic and geometric techniques, which could aid independent verification and suggest new directions for constructing extremal point sets.

major comments (2)
  1. The manuscript does not supply the explicit point coordinates, the algebraic relations enforcing unit lengths, or the incidence count that establishes the excess over the O(n^{4/3}) bound. This information is load-bearing for the central claim of a counterexample; without it, the human-verification assertion cannot be checked directly.
  2. The connections to Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna are asserted in the abstract and reflections but are not accompanied by specific theorem citations or a step-by-step mapping showing how those results are adapted to produce the finite point set.
minor comments (2)
  1. State the precise value of n and the exact number of unit distances achieved so that the violation of the conjectured bound can be quantified.
  2. Add a short appendix or diagram clarifying the geometric configuration if the main text is kept deliberately concise.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on the manuscript. We respond to each major comment below and indicate the planned revisions.

read point-by-point responses

  1. Referee: The manuscript does not supply the explicit point coordinates, the algebraic relations enforcing unit lengths, or the incidence count that establishes the excess over the O(n^{4/3}) bound. This information is load-bearing for the central claim of a counterexample; without it, the human-verification assertion cannot be checked directly.

    Authors: We agree that these details are essential for direct verification of the counterexample claim. The present manuscript is a concise, digested overview emphasizing reflections and conceptual links rather than exhaustive computational data. In the revision we will add an appendix containing the explicit coordinates of the finite point set, the algebraic relations (including minimal polynomials) that enforce the unit distances, and a direct count or comparison establishing that the number of unit distances exceeds the O(n^{4/3}) bound. revision: yes

  2. Referee: The connections to Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna are asserted in the abstract and reflections but are not accompanied by specific theorem citations or a step-by-step mapping showing how those results are adapted to produce the finite point set.

    Authors: We acknowledge that the connections are currently stated at a high level. We will revise the manuscript to include precise citations to the relevant theorems in each of these works and insert a short step-by-step outline in the reflections section that maps the key ideas (e.g., algebraic geometry techniques, pro-p group constructions, and class-field-tower methods) onto the steps used to generate the finite point set. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: counterexample is externally generated and human-verified

full rationale

The paper presents a digested, human-verified version of an OpenAI-generated counterexample rather than deriving the result from first principles within its own text. Key ideas are attributed to independent prior works by Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna. No load-bearing steps reduce by construction to self-definitions, fitted inputs renamed as predictions, or self-citation chains. The central claim rests on the external construction's correctness and verification, which is not shown to loop back to the paper's own inputs.

Assumptions & free parameters 0 free parameters · 1 assumptions · 0 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. No free parameters, invented entities, or non-standard axioms are mentioned in the provided text.

assumptions (1)
  • standard math Standard results and techniques from algebraic number theory and algebraic geometry as developed in the cited works of Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna.
    The abstract states that the argument relies crucially on ideas attributable to these prior sources.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Remarks on the disproof of the unit distance conjecture." pith.science (2026). https://pith.science/paper/HTMKJTNB

@misc{pith2026260520695,
  author       = {Pith},
  title        = {Pith review of: Remarks on the disproof of the unit distance conjecture},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HTMKJTNB}},
  note         = {Machine review of arXiv:2605.20695}
}
read the original abstract

We present a short, digested, human-verified version of the recent OpenAI-generated counterexample to the Erd\H{o}s unit distance conjecture, and a sequence of reflections on it. The argument relies crucially on ideas that may, at least in retrospect, be attributed to Ellenberg-Venkatesh, Golod-Shafarevich, and Hajir-Maire-Ramakrishna.

Discussion (0). Continue with ORCID to comment.

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/AlexanderDuality.lean alexander_duality_circle_linking unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    Lemma 2.1 … projection of Λ onto one of the coordinates … gives a point set P in the plane with 2ν(P) ≥ … and |P| ≤ …

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.

Forward citations

Cited by 27 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. The sum-product conjecture is false for real numbers

    math.NT 2026-05 unverdicted novelty 9.0 of 10

    The sum-product conjecture is false for real numbers, disproved via constructions of sets A with max(|A+A|, |AA|) ≤ |A|^{2-c} for absolute c > 0.

  2. The Erd\H{o}s distinct distances problem in $\mathbb{R}^3$

    math.CO 2026-08 conditional novelty 8.0 of 10

    Any set of N points in R^3 determines at least N^{2/3-o(1)} distinct pairwise distances, matching the lattice construction up to a subpolynomial factor.

  3. The sharp exponent for the minimal distance problem

    math.CO 2026-07 accept novelty 8.0 of 10

    The minimal distance problem has sharp exponent 2/3: lower-bound configurations of size n with separation n^{-2/3−o(1)} are constructed via trace-zero lattices in high-degree totally real number fields.

  4. Split primes and the Elekes-R\'onyai problem

    math.NT 2026-06 unverdicted novelty 8.0 of 10

    Constructs sets A subset R with |{x+y+(x-y)^2 : x,y in A}| <= |A|^{2-c} for some c>0, giving a counterexample to the Elekes-Rónyai problem via prime-splitting amplification.

  5. The number of regular simplices in higher dimensions

    math.CO 2025-07 reject novelty 8.0 of 10

    The paper claims exact maximum counts of equilateral triangles in even dimensions, resolving an Erdős conjecture, but a key proof step is flawed.

  6. The Honeycomb Framework for Code Bounds

    cs.IT 2026-08 conditional novelty 7.0 of 10

    A new hierarchy of representation-theoretic certificates gives provably stronger asymptotic upper bounds on binary code rates than the best previous bounds, including a strict improvement over the 2MQC bound throughou...

  7. Exact certification of the coordinate fields of the triangle-free Exoo-Ismailescu unit-distance graphs EI17 and EI19 (HoG 51375, 51376): a solvable-non-solvable dichotomy (origami vs. S20) and the Laman-number conjecture

    math.CO 2026-07 conditional novelty 7.0 of 10

    The coordinate field of EI17 has irreducible degree 20 and Galois group S20, so its faithful planar realization is not solvable by radicals; EI19 is origami-constructible.

  8. A ChatGPT-assisted Triangle Characterization of Affine Permutation Inversion Graphs

    math.CO 2026-07 accept novelty 7.0 of 10

    Weighted tournaments satisfying the zero-weight condition and Boolean triangle condition on shifted edge weights are exactly the affine inversion graphs.

  9. The Minkowski grid has robustly many repeated distances

    math.CO 2026-07 conditional novelty 7.0 of 10

    There exist n-point planar sets where every subset A has a distance occurring ≥|A|²/n^{1−δ} times, confirming Erdős's 1980 isosceles-triangle conjecture and answering a repeated-distance question negatively.

  10. Rectangles, triangles and Schr\"{o}dinger waves

    math.CA 2026-06 unverdicted novelty 7.0 of 10

    Constructs lattice point sets with many rectangles and few isosceles triangles to produce explicit counterexamples to the Mizohata-Takeuchi conjecture for the paraboloid via transference principles.

  11. Does My Embedding Reflect That $A = B$? Evaluating Mathematical Equivalence in Embedding Models

    cs.CL 2026-06 unverdicted novelty 7.0 of 10

    Embedding models fail to capture mathematical equivalence due to lexical bias, but contrastive training on informal-formal pairs improves performance on MELD and retrieval tasks.

  12. An automated proof that R(B_8,B_10)=37

    math.CO 2026-06 accept novelty 7.0 of 10 full

    Proves R(B_8, B_10) = 37 via an AI-assisted short proof with a Lean formalization of the upper bound.

  13. Can AI Follow In Einstein's Footsteps?

    physics.hist-ph 2026-07 conditional novelty 6.0 of 10

    AI for physics has moved from explicit equation discovery to black-box prediction, a trajectory the authors argue reverses the historical progression of human physics, leaving the invention of new mathematical framewo...

  14. Autonomous disproofs of the sum-product conjecture over $\mathbb R$ with GPT-5.5 Pro

    cs.AI 2026-07 conditional novelty 6.0 of 10

    A three-prompt agent on GPT-5.5 Pro autonomously generated seven apparently correct proofs of the sum-product disproof over R, including two unit-free proofs, at about 132k reasoning tokens per trial.

  15. Testing Frontier Large Language Models' Physics Literacy in Parallel Physical Worlds

    cs.LG 2026-06 unverdicted novelty 6.0 of 10

    Introduces an auditable four-stage diagnostic for LLM physics reasoning in novel frameworks and applies it to three parallel worlds, yielding pass rates of 6/15, 6/15, and 0/15 on frontier models with noted qualitativ...

  16. Rethinking Collaborative Trust for Verifiably Decentralized Blockchain Systems

    cs.CR 2026-06 unverdicted novelty 6.0 of 10

    A framework incentivizes inter-coalition collaboration via asymmetric Shapley values and expander graphs to achieve verifiable decentralization in blockchains.

  17. More sum-product type counterexamples: products with shifts and $AA+A$

    math.NT 2026-06 unverdicted novelty 6.0 of 10

    There exist arbitrarily large finite sets A of reals such that max{|AA|, |(A+1)(A+1)|} ≪ |A|^{2-c} for some c>0.

  18. An Enigma of Artificial Reason: Investigating the Production-Evaluation Gap in Large Reasoning Models

    cs.AI 2026-05 conditional novelty 6.0 of 10

    LRMs show a large production-evaluation gap on the VAIR dataset with valid answers but invalid reasoning, driven by answer confirmation bias as evidenced by CoT analysis, linear probes, and causal patching.

  19. Pre-Strings Lectures on Artificial Intelligence

    hep-th 2026-07 accept novelty 5.5 of 10

    Lecture notes define neural-network field theory and survey how it recovers known QFT/string results plus applied AI techniques for string problems.

  20. Automation Without Understanding

    math.HO 2026-07 conditional novelty 5.0 of 10

    The US is dismantling its mathematical training pipeline precisely as AI begins producing genuine mathematical discoveries, creating a strategic vulnerability that requires policy intervention and formal verification ...

  21. Counterexample to a conjecture on the pairwise independent correlation gap using AI

    math.OC 2026-06 unverdicted novelty 5.0 of 10

    A counterexample is provided to the pairwise independent correlation gap conjecture using AI assistance.

  22. Recursions for Mock Theta Functions

    math.NT 2026-06 unverdicted novelty 5.0 of 10

    Derives weighted recursions for coefficients of mock theta functions f and ω via holomorphic projection on vector-valued Rankin-Cohen brackets, exploiting vanishing cusp form spaces.

  23. Stochastically evolving ellipsoids with symmetries

    math.MG 2026-06 unverdicted novelty 5.0 of 10

    Proves lattice sphere packings in R^N achieve density at least c N² loglog N 2^{-N} for infinitely many N, improving Klartag's bound via stochastic ellipsoid evolution with cyclotomic symmetries.

  24. Optimizing Explicit Unit-Distance Lower-Bound Certificates

    math.OC 2026-06 unverdicted novelty 4.0 of 10

    Optimizes integer multiplicities and a real parameter in Sawin's unit-distance certificate via an evolution strategy to obtain improved explicit lower bounds u(n) > n^{1.0152} and u(n) > n^{1.031}.

  25. How AI settled the complexity of the oldest SGD algorithm

    cs.LG 2026-06 unverdicted novelty 3.0 of 10

    AI models discovered the worst-case complexity of the Kaczmarz algorithm for solving linear systems.

  26. What Remains Human in Mathematics in the Age of AI

    math.HO 2026-07 unverdicted novelty 2.0 of 10

    As AI takes over routine mathematics, the most valuable human contributions shift to problem framing, creative judgment, and genuine understanding.

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

    math.GM 2026-05 unverdicted novelty 2.0 of 10

    An incomplete attempt to show that the O(n^{4/3}) upper bound for unit distances among n points in the plane is not sharp, plus remarks on Szemerédi-Trotter incidences.

Reference graph

Works this paper leans on

37 extracted references · 37 canonical work pages · cited by 27 Pith papers

  1. [1]

    Shabnam Akhtari, Jeffrey D. Vaaler, and Martin Widmer,Effective equidistribution of norm one elements in CM-fields, arXiv preprint arXiv:2507.10387, to appear in Annali della Scuola Normale Superiore di Pisa, Classe di Scienze (2025). 2

    work page arXiv 2025

  2. [2]

    Noga Alon, Matija Bucić, and Lisa Sauermann,Unit and distinct distances in typical norms, Geom. Funct. Anal. 35(2025), 1–42. 1, 16

    work page 2025

  3. [3]

    Marcus Appleby, Steven Flammia, Gary McConnell, and Jon Yard,Generating ray class fields of real quadratic fields via complex equiangular lines, Acta Arithmetica192(2020), 211–233. 2

    work page 2020

  4. [4]

    Batyrev and Yuri Tschinkel,Manin’s conjecture for toric varieties, J

    Victor V. Batyrev and Yuri Tschinkel,Manin’s conjecture for toric varieties, J. Algebraic Geom.7(1998), 15–53. 2 17

    work page 1998

  5. [5]

    Mikhail Belolipetsky and Alexander Lubotzky,Manifolds counting and class field towers, Advances in Mathe- matics229(2012), 3123–3146. 2

    work page 2012

  6. [6]

    Bloom,Erdős problem #90,https://www.erdosproblems.com/90, 2026, Accessed 2026-05-09

    Thomas F. Bloom,Erdős problem #90,https://www.erdosproblems.com/90, 2026, Accessed 2026-05-09. 1

    work page 2026

  7. [7]

    Armand Borel and Gopal Prasad,Finiteness theorems for discrete subgroups of bounded covolume in semi-simple groups, Publications Math’ematiques de l’IH’ES69(1989), 119–171. 5

    work page 1989

  8. [8]

    Peter Brass, William Moser, and János Pach,Research problems in discrete geometry, Springer, New York, 2005. 7

    work page 2005

Show all 37 references
  1. [9]

    Peter Brass, William O. J. Moser, and János Pach,Research problems in discrete geometry, Springer, New York,

  2. [10]

    ID rnm002, 18 pp

    JordanS.EllenbergandAkshayVenkatesh,Reflection principles and bounds for class group torsion, International Mathematics Research Notices2007(2007), Art. ID rnm002, 18 pp. 2, 3

    2007

  3. [11]

    Paul Erdős,Some of my favourite problems which recently have been solved, Proceedings of the International MathematicalConference, Singapore1981(Singapore, 1981), North-HollandMath.Stud., vol.74, North-Holland, Amsterdam-New York, 1982, pp. 59–79. 8

    1981

  4. [12]

    Paul Erdős,Some of my favourite unsolved problems, A tribute to Paul Erdős (1990), 467–478. 8

    1990

  5. [13]

    Paul Erdős,Some of my favourite problems in number theory, combinatorics, and geometry, Resenhas2(1995), 165–186, Combinatorics Week (Portuguese) (SÃo Paulo, 1994). 8

    1995

  6. [14]

    Paul Erdős,On sets of distances ofnpoints, American Mathematical Monthly53(1946), 248–250. 1, 7

    1946

  7. [15]

    E. S. Golod and I. R. Shafarevich,On the class field tower, Izv. Akad. Nauk SSSR Ser. Mat.28(1964), 261–272 (Russian), English translation: Amer. Math. Soc. Transl. (2) 48 (1965), 91–102. 5

    1964

  8. [16]

    E. S. Golod and I. R. Shafarevich,On class field towers, Fourteen Papers on Logic, Algebra, Complex Variables and Topology, American Mathematical Society Translations, Series 2, vol. 48, American Mathematical Society, Providence, RI, 1965, English translation of the 1964 Russi...

    1965

  9. [17]

    Venkatesan Guruswami,Constructions of codes from number fields, IEEE Transactions on Information Theory 49(2003), 594–603. 2

    2003

  10. [18]

    Larry Guth and Nets Hawk Katz,On the Erdős distinct distances problem in the plane, Annals of Mathematics 181(2015), 155–190. 14

    2015

  11. [19]

    II (Barcelona, 2000) (Carles Casacuberta, Rosa Maria Miró-Roig, Joan Verdera, and Sebastià Xambó-Descamps, eds.), Progress in Mathematics, vol

    Farshid Hajir and Christian Maire,Asymptotically good towers of global fields, European Congress of Math- ematics, Vol. II (Barcelona, 2000) (Carles Casacuberta, Rosa Maria Miró-Roig, Joan Verdera, and Sebastià Xambó-Descamps, eds.), Progress in Mathematics, vol. 202, Birkhäus...

    2000

  12. [20]

    2, 3, 5, 6, 16

    Farshid Hajir, Christian Maire, and Ravi Ramakrishna,Cutting towers of number fields, Annales Mathématiques du Québec45(2021), 321–345. 2, 3, 5, 6, 16

    2021

  13. [21]

    Helmut Koch,Galois theory ofp-extensions, Springer Monographs in Mathematics, Springer, Berlin, Heidelberg,

  14. [22]

    Kopp,SIC-POVMs and the Stark conjectures, International Mathematics Research Notices (2021), 13812–13838

    Gene S. Kopp,SIC-POVMs and the Stark conjectures, International Mathematics Research Notices (2021), 13812–13838. 2

    2021

  15. [23]

    Lenstra, H

    Jr. Lenstra, H. W.,Codes from algebraic number fields, Mathematics and Computer Science, II, CWI Mono- graphs, vol. 4, North-Holland Publishing Co., Amsterdam, 1986, Proceedings of the conference held in Amster- dam, 1986, pp. 95–104. 2

    1986

  16. [24]

    Xue-Bin Liang,Theq-ary gilbert-varshamov bound can be improved for all but finitely many positive integersq, 2024, Preprint. 2

    2024

  17. [25]

    S. N. Litsyn and M. A. Tsfasman,Constructive high-dimensional sphere packings, Duke Mathematical Journal 54(1987), 147–161. 2

    1987

  18. [26]

    J. E. Littlewood,On the class-number of the corpusp( √ −k), Proceedings of the London Mathematical Society s2-27(1928), 358–372. 14

    1928

  19. [27]

    Laura Luzzi, Roope Vehkalahti, and Cong Ling,Almost universal codes for MIMO wiretap channels, IEEE Transactions on Information Theory64(2018), 7218–7241. 2

    2018

  20. [28]

    Christian Maire and Frédérique Oggier,Maximal order codes over number fields, Journal of Pure and Applied Algebra222(2018), 1827–1858. 2

    2018

  21. [29]

    James Maynard,Sums of three positive cubes, Journal of the London Mathematical Society113(2026), e70554. 16

    2026

  22. [30]

    Shafarevich,Extensions à points de ramification donnés (en russe), Publications Mathématiques de l’IHÉS18(1963), 71–92 (Russian)

    Igor R. Shafarevich,Extensions à points de ramification donnés (en russe), Publications Mathématiques de l’IHÉS18(1963), 71–92 (Russian). 5

    1963

  23. [31]

    Extensions with prescribed ramification points

    Igor R. Shafarevich,Extensions with given points of ramification, American Mathematical Society Translations, Series 259(1966), 128–149, English translation by J. W. S. Cassels; title also cited as “Extensions with prescribed ramification points”. 5 18

    1966

  24. [32]

    Spencer, Endre Szemerédi, and William T

    Joel H. Spencer, Endre Szemerédi, and William T. Trotter,Unit distances in the euclidean plane, Graph The- ory and Combinatorics (Béla Bollobás, ed.), Academic Press, London, 1984, Proceedings of the Cambridge Conference, 1983, pp. 293–303. 1, 7

    1984

  25. [33]

    Székely,Crossing numbers and hard Erdős problems in discrete geometry, Combinatorics, Probability and Computing6(1997), 353–358

    László A. Székely,Crossing numbers and hard Erdős problems in discrete geometry, Combinatorics, Probability and Computing6(1997), 353–358. 7

    1997

  26. [34]

    Tsfasman,Global fields, codes and sphere packings, Astérisque198–200(1991), 373–396

    Michael A. Tsfasman,Global fields, codes and sphere packings, Astérisque198–200(1991), 373–396. 2

    1991

  27. [35]

    Roope Vehkalahti and Laura Luzzi,Number field lattices achieve gaussian and rayleigh channel capacity within a constant gap, 2015 IEEE International Symposium on Information Theory (ISIT), IEEE, 2015, pp. 436–440. 2

    2015

  28. [36]

    Akshay Venkatesh,A note on sphere packings in high dimension, International Mathematics Research Notices (2013), 1628–1642. 2

    2013

  29. [37]

    Walfisz,On the class-number of binary quadratic forms, Trav. Inst. Math. Tbilissi11(1942), 57–71. 14 Email address:nalon@math.princeton.edu Email address:thomas.bloom@manchester.ac.uk Email address:w.t.gowers@dpmms.cam.ac.uk Email address:daniel.litt@utoronto.ca Email address:...

    1942

Pith tools

Reviewed May 21, 2026 · model on record in the stance chip above.