pith. sign in

arxiv: 2605.28709 · v1 · pith:CHS5Y4KCnew · submitted 2026-05-27 · 🧮 math.CO · math.MG

Improved bounds for the double cap conjecture

Domonkos Czifra , \'Akos D\'ucz , M\'at\'e Matolcsi , D\'aniel Varga , P\'al Zs\'amboki This is my paper

Pith reviewed 2026-06-29 11:25 UTC · model grok-4.3

classification 🧮 math.CO math.MG
keywords double cap conjectureorthogonal vectorsmeasurable densitygeometric fractional chromatic numberspherical point setscomputer searchupper boundsharmonic analysis
0
0 comments X

The pith

A 33-point set on the sphere yields an upper bound of 0.2953 on the maximum density of measurable sets without orthogonal vectors.

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

The paper improves the known upper bound on α_3, the highest measurable density of a subset of the 3-sphere containing no orthogonal pair of vectors, from 0.2977 down to 0.2953. It reaches this figure by computing the geometric fractional chromatic number of a 33-element point set located through extensive computer search. This reduces the distance to the conjectured optimum of roughly 0.2929 supplied by the double-cap construction of two opposite caps. Readers care because the result shows that modest finite configurations can produce concrete numerical improvements on a long-standing density question in low dimensions. The same technique is indicated to extend directly to higher-dimensional spheres.

Core claim

By identifying an appropriate 33-element point set through a large-scale computer search, the geometric fractional chromatic number of this set supplies the upper bound α_3 ≤ 0.2953 on the maximum density of a measurable subset of the 3-sphere containing no orthogonal vectors, improving the previous best upper bound of 0.2977.

What carries the argument

The geometric fractional chromatic number of a finite point set on the sphere, which converts a discrete coloring problem into an upper bound on the continuous measurable density α_n through harmonic-analytic arguments.

If this is right

  • The gap between the best lower bound of approximately 0.2929 and the upper bound for α_3 shrinks from 0.0048 to 0.0024.
  • The method applies verbatim in higher dimensions and can produce improved numerical bounds there.
  • Finite point sets of small cardinality suffice to generate explicit, computable upper bounds on α_n.
  • Large-scale search over point configurations on the sphere is a viable route to tighter density estimates.

Where Pith is reading between the lines

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

  • If still smaller point sets can be shown to achieve comparable or better bounds, exhaustive enumeration might eventually become practical.
  • The technique may transfer to other forbidden-configuration problems on spheres where measurable density is the quantity of interest.
  • Further refinement of the search heuristic could close more of the remaining gap to the double-cap value without enlarging the point set.

Load-bearing premise

The geometric fractional chromatic number of any finite point set on the sphere supplies a valid upper bound on the measurable density α_n.

What would settle it

An explicit measurable set on the 3-sphere with density strictly larger than 0.2953 that contains no pair of orthogonal vectors would disprove the claimed bound.

Figures

Figures reproduced from arXiv: 2605.28709 by \'Akos D\'ucz, D\'aniel Varga, Domonkos Czifra, M\'at\'e Matolcsi, P\'al Zs\'amboki.

Figure 1
Figure 1. Figure 1: The 33-vertex graph GX visualized. Left figure slightly elevated viewpoint, right figure orthogonal projection onto the xy-plane. Numbers correspond to row indices in [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
read the original abstract

In 1974, Witsenhausen asked for the maximum possible density $\alpha_n$ of a measurable subset $A$ of the unit sphere $\mathbb{S}^{n-1}\subset \mathbb{R}^n$ such that $A$ contains no pair of orthogonal vectors. For $n=3$, the best known lower bound is $1 - 1/\sqrt{2} = 0.29289\dots$, obtained from the natural "double cap" construction of two opposite spherical caps, which is conjectured to be optimal for all $n$ by Gil Kalai. In this paper, we use a novel approach to establish an upper bound of $\alpha_3\le 0.2953$, improving the previous best known bound $0.2977$ due to Bekker et al. (2025). Our approach combines harmonic-analytic arguments with the geometric fractional chromatic number of finite graphs, recently introduced by Ambrus et al. (2024). In this framework, any finite subset of the sphere yields an upper bound for $\alpha_n$, and we obtain our bound by identifying an appropriate 33-element point set through a large-scale computer search. The same method can also be used in higher dimensions to yield potential improvements of the best known bounds.

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

2 major / 1 minor

Summary. The manuscript claims an improved upper bound α_3 ≤ 0.2953 on the maximum measurable density of a subset of S^2 with no orthogonal pair, improving the prior bound 0.2977. The bound is obtained by applying the geometric fractional chromatic number construction of Ambrus et al. (2024) to a 33-point configuration located by large-scale computer search; the same method is suggested for higher dimensions.

Significance. If the cited framework applies without gap and the numerical computation is reproducible, the result supplies a modest but concrete advance on Witsenhausen's problem and Gil Kalai's double-cap conjecture. The combination of harmonic analysis with the new finite-graph tool and computer-assisted configuration search is a methodological contribution that could be reused.

major comments (2)
  1. [computational search section] The central numerical claim α_3 ≤ 0.2953 rests on the 33-point set and its geometric fractional chromatic number, yet the manuscript provides neither the explicit coordinates of the configuration nor the search algorithm, objective function, or verification that the computed value is exactly 0.2953. This information is load-bearing for the stated improvement.
  2. [Abstract and framework application paragraph] The reduction from measurable density α_n to the geometric fractional chromatic number of a finite point set is invoked via the Ambrus et al. (2024) framework (Abstract). The manuscript does not supply an independent argument or address possible measurability or approximation gaps in this reduction, which is the sole justification for converting the finite-graph quantity into a bound on α_3.
minor comments (1)
  1. [Abstract] The abstract states the bound to four decimal places (0.2953) without indicating the precision of the underlying chromatic-number computation or any rounding convention.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their detailed report and constructive suggestions. We address each major comment below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [computational search section] The central numerical claim α_3 ≤ 0.2953 rests on the 33-point set and its geometric fractional chromatic number, yet the manuscript provides neither the explicit coordinates of the configuration nor the search algorithm, objective function, or verification that the computed value is exactly 0.2953. This information is load-bearing for the stated improvement.

    Authors: We agree that the coordinates, algorithm, objective function, and verification are necessary for full reproducibility of the numerical claim. In the revised manuscript we will add the explicit coordinates of the 33-point set, a description of the large-scale search procedure and objective function employed, and the computational verification confirming that the geometric fractional chromatic number of this configuration produces the bound 0.2953. Supplementary material containing the data and verification code will also be provided. revision: yes

  2. Referee: [Abstract and framework application paragraph] The reduction from measurable density α_n to the geometric fractional chromatic number of a finite point set is invoked via the Ambrus et al. (2024) framework (Abstract). The manuscript does not supply an independent argument or address possible measurability or approximation gaps in this reduction, which is the sole justification for converting the finite-graph quantity into a bound on α_3.

    Authors: The bound relies on the reduction established in the cited Ambrus et al. (2024) framework. We will revise the abstract and the relevant paragraph in the introduction to include a concise summary of the key steps in that reduction, with explicit citations to the theorems in Ambrus et al. that justify applying the geometric fractional chromatic number to measurable densities. The framework is formulated precisely for measurable sets, and our application introduces no further approximation; we will note this explicitly in the revision. revision: yes

Circularity Check

0 steps flagged

No circularity; bound obtained via external framework applied to computer-searched configuration

full rationale

The derivation chain consists of (1) invoking the geometric fractional chromatic number framework introduced in the independent Ambrus et al. (2024) paper, which the present work treats as an external black-box that converts any finite point set into an upper bound on measurable α_n, and (2) using a large-scale computer search to locate a 33-element set that yields the numerical value 0.2953 under that framework. No equation, definition, or claim in the abstract reduces the final bound to a parameter fitted from the target quantity itself, nor does any load-bearing step rest on a self-citation chain whose cited result is unverified. The computer search is a search for an optimizing instance rather than a renaming or self-definition of the output. The paper is therefore self-contained against external benchmarks and receives the default non-circularity finding.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the validity of the geometric fractional chromatic number framework (Ambrus et al. 2024) and the correctness of the computer search identifying a 33-point configuration that yields the stated bound.

axioms (1)
  • domain assumption Any finite subset of the sphere yields a valid upper bound on the measurable density α_n via its geometric fractional chromatic number
    This is the key framework invoked to convert the finite configuration into the continuous bound.

pith-pipeline@v0.9.1-grok · 5782 in / 1173 out tokens · 30071 ms · 2026-06-29T11:25:33.606849+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

17 extracted references · 3 canonical work pages

  1. [1]

    Ambrus, A

    G. Ambrus, A. Csisz´ arik, M. Matolcsi, D. Varga, and P. Zs´ amboki, The density of planar sets avoiding unit distances, Math. Program., 207 (2024), pp. 303-327,

    2024

  2. [2]

    Vera, Optimization hierarchies for distance-avoiding sets in compact spaces, Trans

    Bram Bekker, Olga Kuryatnikova, Fernando M´ ario de Oliveira Filho, and Juan C. Vera, Optimization hierarchies for distance-avoiding sets in compact spaces, Trans. Amer. Math. Soc. 379 (2026), no. 1, pp. 33-70,https://doi.org/10. 1090/tran/9578

    2026

  3. [3]

    Coulom, R. (2007). Efficient Selectivity and Backup Operators in Monte-Carlo Tree Search. In: van den Herik, H.J., Ciancarini, P., Donkers, H.H.L.M. (eds) Computers and Games 2006. Lecture Notes in Computer Science, vol 4630. Springer, Berlin, Heidelberg.https://doi.org/10.1007/978-3-540-75538-8_7

    work page doi:10.1007/978-3-540-75538-8_7 2007

  4. [4]

    DeCorte, F.M

    E. DeCorte, F.M. de Oliveira Filho, and F. Vallentin, Complete positivity and distance- avoiding sets, Mathematical Programming, Series A 191 (2022) 487-558

    2022

  5. [5]

    DeCorte, O

    E. DeCorte, O. Pikhurko: Spherical Sets Avoiding a Prescribed Set of Angles, International Mathematics Research Notices, Volume 2016, Issue 20, 2016, Pages 6095-6117,

    2016

  6. [6]

    Dai and Y

    F. Dai and Y. Xu: Approximation theory and harmonic analysis on spheres and balls. Springer Monographs in Mathe- matics. Springer, Berlin, 2013

    2013

  7. [7]

    Frankl and R

    P. Frankl and R. M. Wilson: Intersection theorems with geometric consequences. Combinatorica, 1(4):357-368, 1981

    1981

  8. [8]

    Available athttps://www.gurobi.com, 2025

    Gurobi Optimization, LLC.Gurobi Optimizer Reference Manual. Available athttps://www.gurobi.com, 2025

    2025

  9. [9]

    G. Kalai: Some old and new problems in combinatorial geometry I: around Borsuk’s problem, in: Surveys in combina- torics 2015, London Mathematical Society Lecture Note Series 424, Cambridge University Press, Cambridge, 2015, pp. 147-174

    2015

  10. [10]

    Bandit Based Monte-Carlo Planning

    Kocsis, L., Szepesv´ ari, C. Bandit Based Monte-Carlo Planning. In: F¨ urnkranz, J., Scheffer, T., Spiliopoulou, M. (eds) Machine Learning: ECML 2006. Lecture Notes in Computer Science(), vol 4212. Springer, Berlin, Heidelberg.https: //doi.org/10.1007/11871842_29

    work page doi:10.1007/11871842_29 2006

  11. [11]

    Matolcsi, I

    M. Matolcsi, I. Ruzsa, D. Varga, P. Zs´ amboki: The fractional chromatic number of the plane is at least 4

  12. [12]

    A. M. Raigorodskii: On a bound in Borsuk’s problem. Russian Mathematical Surveys, 54(2):453-454, 1999

    1999

  13. [13]

    I. J. Schoenberg: Metric spaces and positive definite functions. Trans. Amer. Math. Soc., 44:522-536, 1938

    1938

  14. [14]

    Silver, D., Schrittwieser, J., Simonyan, K. et al. Mastering the game of Go without human knowledge. Nature 550, 354–359 (2017).https://doi.org/10.1038/nature24270

    work page doi:10.1038/nature24270 2017

  15. [15]

    Szeg˝ o, Orthogonal Polynomials (Fourth Edition), American Mathematical Society Colloquium Publications, Volume XXIII, American Mathematical Society, Providence, 1975

    G. Szeg˝ o, Orthogonal Polynomials (Fourth Edition), American Mathematical Society Colloquium Publications, Volume XXIII, American Mathematical Society, Providence, 1975

    1975

  16. [16]

    H. S. Witsenhausen: Spherical sets without orthogonal point pairs. American Mathematical Monthly, pages 1101-1102, 1974

    1974

  17. [17]

    Available athttp://bit.ly/witsenhausen

    Supplementary material for this paper (verification code and data). Available athttp://bit.ly/witsenhausen. 14 Czifra Domonkos, HUN-REN Alfr´ed R´enyi Institute of Mathematics, Re´altanoda utca 13-15, 1053 Budapest, Hungary Email address:czifra.domonkos@renyi.hu ´Akos D´ucz, HUN-REN Alfr´ed R´enyi Institute of Mathematics, Re´altanoda utca 13-15, 1053 Bud...