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arxiv: 2606.03987 · v1 · pith:5QIEKLFBnew · submitted 2026-06-02 · 🧮 math.CO · math.MG

Kusner's conjecture: Exact values and linear bounds

Hong-Jun Ge , Zixiang Xu , Yang Zhou This is my paper

Pith reviewed 2026-06-28 09:17 UTC · model grok-4.3

classification 🧮 math.CO math.MG
keywords Kusner's conjectureequilateral setsl_p metriclinear boundstoruscombinatorics
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The pith

The largest equilateral set in R^n with the l_p metric has size at most (2k+1)n+1 for p in [4k+2,4k+4].

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

The paper proves Kusner's conjecture that the largest equilateral set in R^n under the l_p metric has cardinality n+1 when 2 ≤ p ≤ 4. It generalizes the result to show that the size is at most (2k+1)n+1 for p in each interval [4k+2,4k+4]. Almost linear bounds of order n log n are given for the complementary intervals (4k,4k+2). Similar improved bounds are derived for equilateral sets on the torus T^n.

Core claim

We prove Kusner's conjecture for every dimension n≥1 when 2≤p≤4. More generally, for every integer k≥0 and every p∈[4k+2,4k+4], every equilateral set in R^n with metric ℓ_p has cardinality at most (2k+1)n+1. On the complementary intervals p∈(4k,4k+2) with p≥1, we obtain the almost linear bound O_p(n log n). We also prove the almost linear bound O_p(n log n) for 1≤p≤2 and O_p(n^{3/2-1/p}) for p>2 on the torus T^n.

What carries the argument

Convexity and differentiability properties of the ℓ_p norm that translate the equilateral condition into linear or combinatorial constraints.

Load-bearing premise

The convexity and differentiability properties of the l_p norm in those intervals translate the equilateral condition into a system of linear or combinatorial constraints that cannot be satisfied by more than (2k+1)n+1 points.

What would settle it

An explicit construction of more than (2k+1)n+1 equilateral points in R^n for some p in [4k+2,4k+4] would disprove the upper bound.

read the original abstract

In 1983, Kusner conjectured that the largest equilateral set in $\mathbb{R}^{n}$ with metric $\ell_{p}$ has cardinality $n+1$ when $1<p<\infty$ and $2n$ when $p=1.$ This conjecture was proved only in the isolated cases $p=2$ and $p=4$, and was disproved when $1<p<2$. The best general upper bound $O_p(n^{\frac{2p+2}{2p-1}})$ is due to the celebrated work of Alon and Pudl\'ak~[GAFA, 2003]. Our main contributions include: (1) We prove Kusner's conjecture for every dimension $n\ge 1$ when $2\le p\le 4$. More generally, for every integer $k\ge 0$ and every $p\in[4k+2,4k+4]$, every equilateral set in \(\mathbb{R}^{n}\) with metric $\ell_p$ has cardinality at most $(2k+1)n+1$. On the complementary intervals $p\in(4k,4k+2)$ with $p\geq 1$, we obtain the almost linear bound $O_p(n\log n)$. (2) We also consider the analogous problem on the torus $\mathbb{T}^n$, recently initiated by Alon, where the cyclic distance makes the problem substantially more delicate than in $\mathbb R^n$. We prove the almost linear bound $O_p(n\log n)$ for $1\le p\le 2$ and $O_p(n^{\frac{3}{2}-\frac{1}{p}})$ for every fixed real $p>2$, improving Alon's bounds $O_p(n^{2+\frac{2}{\lfloor p\rfloor}})$ for all finite $p\ge 1$.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript proves Kusner's 1983 conjecture on the maximum size of equilateral sets in R^n under the l_p metric, establishing the bound n+1 for all n≥1 when 2≤p≤4. More generally, it shows that for each integer k≥0 and p∈[4k+2,4k+4] the cardinality is at most (2k+1)n+1; on the complementary intervals p∈(4k,4k+2) it obtains the almost-linear bound O_p(n log n). Analogous results are proved for the torus T^n, improving Alon's earlier bounds to O_p(n log n) for 1≤p≤2 and O_p(n^{3/2-1/p}) for p>2.

Significance. If the central arguments hold, the work resolves the conjecture on a positive-measure set of p-values (including the previously open interval (2,4)) and replaces the Alon-Pudlák superlinear bound with linear or near-linear estimates in wide ranges of p. The torus results constitute a clear improvement over the state of the art. The paper supplies explicit linear bounds rather than asymptotic statements and builds directly on the cited external reference for the torus case.

major comments (1)
  1. [Abstract / main contributions (1)] Abstract, paragraph on main contributions (1), and the proof of the linear bound for p∈[4k+2,4k+4]: the reduction of the system {‖x_i−x_j‖_p=1}_{i≠j} to a collection of linear inequalities or combinatorial obstructions whose only solutions have size ≤(2k+1)n+1 rests on specific convexity and higher-order differentiability properties of the l_p norm on those closed intervals. The manuscript must exhibit the explicit Taylor expansion, subdifferential description, or convexity argument that produces the obstruction; any gap in the p-dependent analysis away from the endpoints 4k+2 and 4k+4 would invalidate the cardinality claim.
minor comments (2)
  1. [Introduction] The distinction between the closed intervals [4k+2,4k+4] and the open complementary intervals (4k,4k+2) should be stated uniformly in the introduction and in the statement of the torus theorems to prevent notational ambiguity.
  2. [Introduction] A short table or explicit list of the recovered cases (p=2, p=4, k=0) would help readers verify consistency with the known results cited in the abstract.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for highlighting the need for greater explicitness in the convexity arguments. We will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract / main contributions (1)] Abstract, paragraph on main contributions (1), and the proof of the linear bound for p∈[4k+2,4k+4]: the reduction of the system {‖x_i−x_j‖_p=1}_{i≠j} to a collection of linear inequalities or combinatorial obstructions whose only solutions have size ≤(2k+1)n+1 rests on specific convexity and higher-order differentiability properties of the l_p norm on those closed intervals. The manuscript must exhibit the explicit Taylor expansion, subdifferential description, or convexity argument that produces the obstruction; any gap in the p-dependent analysis away from the endpoints 4k+2 and 4k+4 would invalidate the cardinality claim.

    Authors: The proof in Section 3 proceeds by showing that for p ∈ [4k+2, 4k+4] the map x ↦ ‖x‖_p^p is strictly convex, with the second derivative (or appropriate subdifferential when p is even) yielding a positive-definite quadratic form that forces any equilateral set to satisfy a system of linear inequalities whose only solutions have cardinality at most (2k+1)n+1. We acknowledge that the current write-up sketches this reduction without spelling out the Taylor expansion or the explicit subdifferential description at every point of the interval. In the revision we will insert a dedicated subsection that (i) states the Taylor expansion of |t|^p up to second order for t near the coordinate differences, (ii) describes the subdifferential when p is an even integer, and (iii) verifies that the resulting obstruction is uniform on the closed interval [4k+2, 4k+4] and does not degenerate away from the endpoints. This addition will make the argument self-contained. revision: yes

Circularity Check

0 steps flagged

No circularity: new proofs via norm convexity yield independent bounds

full rationale

The paper derives its main results (exact Kusner conjecture for 2≤p≤4 and linear bounds (2k+1)n+1 for p∈[4k+2,4k+4]) by translating the equilateral condition ||x_i−x_j||_p=1 into linear/combinatorial constraints using the convexity and differentiability properties of the ℓ_p norm on those intervals. This reduction is presented as a direct mathematical argument rather than a self-definition, a fitted parameter renamed as a prediction, or a load-bearing self-citation. The torus bounds improve an external reference (Alon) without reducing to it by construction. No quoted step equates a claimed output to its input by definition or statistical forcing, so the derivation chain remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work operates inside the standard axiomatic framework of real analysis, normed spaces, and finite combinatorics; no free parameters are fitted to data and no new entities are postulated.

axioms (1)
  • standard math Standard properties of the ℓ_p norm (triangle inequality, convexity for p≥1) and the definition of an equilateral set in a metric space
    Invoked throughout the statements of the conjecture and the new bounds.

pith-pipeline@v0.9.1-grok · 5877 in / 1598 out tokens · 45479 ms · 2026-06-28T09:17:46.575479+00:00 · methodology

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Reference graph

Works this paper leans on

21 extracted references

  1. [1]

    N. Alon. Equilateral sets in the torus. https://web.math.princeton.edu/~nalon/PDFS/ equitor.pdf, 2026

    2026

  2. [2]

    Alon and P

    N. Alon and P. Pudl´ ak. Equilateral sets inl n p .Geom. Funct. Anal., 13(3):467–482, 2003

    2003

  3. [3]

    Balla, F

    I. Balla, F. Dr¨ axler, P. Keevash, and B. Sudakov. Equiangular lines and spherical codes in Euclidean space.Invent. Math., 211(1):179–212, 2018

    2018

  4. [4]

    Bandelt, V

    H.-J. Bandelt, V. Chepoi, and M. Laurent. Embedding into rectilinear spaces.Discrete Comput. Geom., 19(4):595–604, 1998

    1998

  5. [5]

    Bilyk, N

    D. Bilyk, N. Nagel, and I. Ruohoniemi. Minimizing point configurations for tensor product energies on the torus.arXiv preprint, arXiv: 2510.25442, 2025

    arXiv 2025

  6. [6]

    B. Bukh. Bounds on equiangular lines and on related spherical codes.SIAM J. Discrete Math., 30(1):549–554, 2016

    2016

  7. [7]

    R. Chen, F. Gui, J. Tang, and N. Xiong. Few distance sets in ℓp spaces and ℓp product spaces. European J. Combin., 102:Paper No. 103459, 14, 2022

    2022

  8. [8]

    Gershgorin

    S. Gershgorin. ¨Uber die Abgrenzung der Eigenwerte einer Matrix.Bull. Acad. Sci. URSS, 1931(6):749–754, 1931

    1931

  9. [9]

    R. K. Guy. Unsolved Problems: An Olla-Podrida of Open Problems, Often Oddly Posed.Amer. Math. Monthly, 90(3):196–200, 1983

    1983

  10. [10]

    Jiang, J

    Z. Jiang, J. Tidor, Y. Yao, S. Zhang, and Y. Zhao. Equiangular lines with a fixed angle.Ann. of Math. (2), 194(3):729–743, 2021

    2021

  11. [11]

    S. V. Konyagin. On systems of vectors with equal distances.Trudy Matematicheskogo Tsentra imeni N. I. Lobachevskogo, 43:204–209, 2011. In Russian; English translation available

    2011

  12. [12]

    Koolen, M

    J. Koolen, M. Laurent, and A. Schrijver. Equilateral dimension of the rectilinear space. volume 21, pages 149–164. 2000. Special issue dedicated to Dr. Jaap Seidel on the occasion of his 80th birthday (Oisterwijk, 1999)

    2000

  13. [13]

    P. W. H. Lemmens and J. J. Seidel. Equiangular lines.J. Algebra, 24:494–512, 1973

    1973

  14. [14]

    C. M. Petty. Equilateral sets in Minkowski spaces.Proc. Amer. Math. Soc., 29:369–374, 1971

    1971

  15. [15]

    Roman.Advanced linear algebra, volume 135 ofGraduate Texts in Mathematics

    S. Roman.Advanced linear algebra, volume 135 ofGraduate Texts in Mathematics. Springer-Verlag, New York, 1992

    1992

  16. [16]

    Smyth.The conjectures of Rudich, Tardos, and Kusner

    C. Smyth.The conjectures of Rudich, Tardos, and Kusner. ProQuest LLC, Ann Arbor, MI, 2001. Thesis (Ph.D.)–Rutgers The State University of New Jersey - New Brunswick

    2001

  17. [17]

    C. Smyth. Equilateral sets in ℓd p. In J. Pach, editor,Thirty Essays on Geometric Graph Theory, pages 483–487. Springer, New York, 2013

    2013

  18. [18]

    K. J. Swanepoel. A problem of Kusner on equilateral sets.Arch. Math. (Basel), 83(2):164–170, 2004

    2004

  19. [19]

    K. J. Swanepoel. Equilateral sets and a Sch¨ utte theorem for the 4-norm.Canad. Math. Bull., 57(3):640–647, 2014. 46

    2014

  20. [20]

    J. H. van Lint and J. J. Seidel. Equilateral point sets in elliptic geometry.Indag. Math., 28:335–348,

  21. [21]

    Nederl. Akad. Wetensch. Proc. Ser. A 69. 47