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arxiv: 2510.17453 · v2 · submitted 2025-10-20 · 🧮 math.CA · math.CO

Joint upper Banach density, VC dimensions and Euclidean point configurations

Bruno Predojevi\'c This is my paper

Pith reviewed 2026-05-18 06:21 UTC · model grok-4.3

classification 🧮 math.CA math.CO
keywords joint upper Banach densityVC dimensionconvex curvespoint configurationsEuclidean distancesmeasurable setsVapnik-Chervonenkis dimension
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The pith

Joint upper Banach density for two sets ensures realization of all large distances along a convex curve and maximal VC dimension for scaled curve families.

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

The paper defines two generalizations of upper Banach density applicable to pairs of measurable subsets of the plane. The first of these is used to extend a classic theorem about large distances in a single positive-density set to distances realized between points from two different sets under a suitable joint density condition. The second generalization is applied to prove that, for all sufficiently large scales t, the family of portions of translates of tΓ, where Γ is a smooth closed centrally symmetric planar curve bounding a convex compact region of non-vanishing curvature, attains the maximal possible Vapnik-Chervonenkis dimension. Sympathetic readers would care as these links between density, metric geometry, and combinatorial dimension theory provide new ways to analyze which point configurations must appear in sufficiently dense subsets of the plane.

Core claim

We study joint upper Banach densities of two sets in the plane. This allows us to generalize results on large distances realized in positive upper density sets to the setting of two sets satisfying an appropriate density condition. For the second quantity, we show that the family of portions of translates of tΓ has the maximal possible VC dimension for all sufficiently large t > 0, when Γ is a smooth, closed, centrally symmetric planar curve bounding a convex compact region of non-vanishing curvature.

What carries the argument

Joint upper Banach density of a pair of measurable sets in the plane, which controls the simultaneous appearance of points from each set in large regions and enables both the distance generalization and the VC-dimension bound.

If this is right

  • If two sets satisfy the joint upper Banach density condition, their difference set contains all sufficiently large points lying on any translate of the given curve Γ.
  • The family of arc portions from large translates of Γ achieves the highest attainable VC dimension.
  • This provides a bipartite version of classical single-set distance theorems in the plane.
  • Such results constrain the possible avoidance of certain geometric configurations in pairs of dense measurable sets.

Where Pith is reading between the lines

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

  • The approach might extend to studying joint densities in higher-dimensional spaces or on manifolds.
  • Concrete examples with specific curves like the circle could be computed to illustrate the VC-dimension result.
  • These density notions could connect to problems in discrete geometry involving forbidden configurations.

Load-bearing premise

The curve Γ must be smooth with non-vanishing curvature, closed, centrally symmetric and bound a convex compact set, while the two sets must satisfy the stated joint density condition.

What would settle it

A counterexample would consist of a pair of sets with positive joint upper Banach density whose difference avoids all sufficiently large points on some qualifying curve Γ, or a calculation showing that the VC dimension of the translated arc family falls below the maximum for some large t.

Figures

Figures reproduced from arXiv: 2510.17453 by Bruno Predojevi\'c.

Figure 1
Figure 1. Figure 1: Flexible configuration from Theorem 1.7 Another consequence of Proposition 3.2 below tells us that for a measurable set E ⊆ R2 , we have the chain of equivalences δ(E) > 0 ⇐⇒ δV C(E, E) > 0 ⇐⇒ δV C(R2 , E) > 0 ⇐⇒ δV C(E, R2 ) > 0, which tells us that the qualitative assumption of quantity from Definition 1.5 being strictly positive for A = B = E is equivalent to the condition of E being a set of strictly p… view at source ↗
Figure 2
Figure 2. Figure 2: Sketch of proof of Lemma 2.5 Proof. By making an appropriate change of coordinates, the claim reduces to checking 3 distinct cases which correspond to different arrangements of the given points and which are illustrated in [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
read the original abstract

We study two related quantities which generalize the concept of upper Banach density of a set to two measurable subsets of the plane. The first of them allows us to generalize a classic result on sufficiently large distances realized in a set of positive upper density, to distances between points of two sets satisfying an appropriate density condition. The second one allows us to show that for all sufficiently large scales $t>0$ and for a smooth, closed, centrally symmetric, planar curve $\Gamma$ which bounds a convex and compact region in the plane and is of non-vanishing curvature, the family consisting of portions of translates of $t\Gamma$ has the maximal possible Vapnik--Chervonenkis dimension.

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

0 major / 3 minor

Summary. The manuscript introduces two joint upper Banach density quantities for pairs of measurable subsets of the plane. The first generalizes results on large distances realized in sets of positive upper density to distances between points from two sets satisfying a joint density condition. The second is applied to show that, for a smooth, closed, centrally symmetric planar curve Γ bounding a convex compact region with non-vanishing curvature, the family of portions of translates of tΓ attains the maximal possible VC-dimension for all sufficiently large t > 0.

Significance. If the central claims are established rigorously, the work provides a bridge between density theory in the plane and VC-dimension theory for families of curves. The distance result extends classical theorems in geometric measure theory, while the VC-dimension result offers a concrete application to point configurations avoiding or realizing certain geometric patterns. The assumptions on Γ are standard and ensure the curve behaves like a strictly convex oval, allowing the VC-dimension to be bounded above while the density condition provides the shattering lower bound.

minor comments (3)
  1. The abstract states that proofs exist for the two main results but provides no outline of the key steps or explicit handling of the non-vanishing curvature assumption; consider adding one sentence summarizing the main ideas of each proof.
  2. Define the two joint upper Banach density quantities with full notation and properties in the introduction or §1 before they appear in the statements of the main theorems.
  3. Clarify whether the joint density condition in the distance result is symmetric in the two sets or requires one to be denser than the other.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript, the positive assessment of its significance, and the recommendation for minor revision. We appreciate the summary provided and will incorporate improvements for clarity in the revised version.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper generalizes classical results on upper Banach density to joint versions for two sets and applies them to prove a VC-dimension bound for families of curve portions under standard external geometric hypotheses (smoothness, central symmetry, convexity, compactness, non-vanishing curvature). These assumptions are stated as inputs rather than derived internally, and the shattering constructions rely on the joint-density hypothesis without reducing the claimed maximal VC dimension back to a fit, self-definition, or self-citation chain. No load-bearing step collapses by the paper's own equations to its inputs; the derivation remains self-contained against external classical results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper introduces two new density definitions and relies on standard assumptions from real analysis and VC theory; no free parameters or invented entities are visible in the abstract.

axioms (1)
  • standard math Standard properties of upper Banach density and VC dimension from prior literature hold for the generalized joint versions.
    Invoked implicitly when generalizing the classic distance result and when claiming maximal VC dimension.

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Works this paper leans on

34 extracted references · 34 canonical work pages

  1. [1]

    Miller, Francisco Romero Acosta, and Santiago Velazquez Iannuzzelli

    Ruben Ascoli, Livia Betti, Justin Cheigh, Alex Iosevich, Ryan Jeong, Xuyan Liu, Brian McDonald, Wyatt Milgrim, Steven J. Miller, Francisco Romero Acosta, and Santiago Velazquez Iannuzzelli. VC-dimension and distance chains inF d q.Korean J. Math., 32(1):43–57, 2024

    work page 2024

  2. [2]

    Miller, Francisco Romero Acosta, and Santiago Velazquez Innuzzelli

    Ruben Ascoli, Livia Betti, Justin Cheigh, Alex Iosevich, Ryan Jeong, Xuyan Liu, Brian McDonald, Wyatt Milgrim, Steven J. Miller, Francisco Romero Acosta, and Santiago Velazquez Innuzzelli. VC-dimension of hyperplanes over finite fields.Graphs Combin., 41(2):Paper No. 47, 13, 2025. doi:10.1007/s00373-025-02909-6

    work page doi:10.1007/s00373-025-02909-6 2025

  3. [3]

    A Szemer´ edi type theorem for sets of positive density inR k.Israel J

    Jean Bourgain. A Szemer´ edi type theorem for sets of positive density inR k.Israel J. Math., 54(3):307–316, 1986.doi:10.1007/BF02764959

    work page doi:10.1007/bf02764959 1986

  4. [4]

    A Roth-type theorem for dense subsets of Rd.Bull

    Brian Cook, ´Akos Magyar, and Malabika Pramanik. A Roth-type theorem for dense subsets of Rd.Bull. Lond. Math. Soc., 49(4):676–689, 2017.doi:10.1112/blms.12043

    work page doi:10.1112/blms.12043 2017

  5. [5]

    do Carmo.Differential Geometry of Curves and Surfaces: Revised and Updated

    Manfredo P. do Carmo.Differential Geometry of Curves and Surfaces: Revised and Updated. Dover Publications, Mineola, NY, revised and updated edition, 2016

    work page 2016

  6. [6]

    An L 4 estimate for a singular entangled quadrilinear form.Math

    Polona Durcik. An L 4 estimate for a singular entangled quadrilinear form.Math. Res. Lett., 22(5):1317–1332, 2015.doi:10.4310/MRL.2015.v22.n5.a3

    work page doi:10.4310/mrl.2015.v22.n5.a3 2015

  7. [7]

    L p estimates for a singular entangled quadrilinear form.Trans

    Polona Durcik. L p estimates for a singular entangled quadrilinear form.Trans. Amer. Math. Soc., 369(10):6935–6951, 2017.doi:10.1090/tran/6850

    work page doi:10.1090/tran/6850 2017

  8. [8]

    Boxes, extended boxes and sets of positive upper density in the Euclidean space.Math

    Polona Durcik and Vjekoslav Kovaˇ c. Boxes, extended boxes and sets of positive upper density in the Euclidean space.Math. Proc. Cambridge Philos. Soc., 171(3):481–501, 2021.doi:10.1017/ S0305004120000316

    work page 2021

  9. [9]

    A Szemer´ edi-type theorem for subsets of the unit cube.Anal

    Polona Durcik and Vjekoslav Kovaˇ c. A Szemer´ edi-type theorem for subsets of the unit cube.Anal. PDE, 15(2):507–549, 2022.doi:10.2140/apde.2022.15.507

    work page doi:10.2140/apde.2022.15.507 2022

  10. [10]

    On side lengths of corners in positive density subsets of the Euclidean space.Int

    Polona Durcik, Vjekoslav Kovaˇ c, and Luka Rimani´ c. On side lengths of corners in positive density subsets of the Euclidean space.Int. Math. Res. Not. IMRN, 14(22):6844–6869, 2018.doi:10. 1093/imrn/rnx093

    work page 2018

  11. [11]

    Local bounds for singular Brascamp- Lieb forms with cubical structure.Math

    Polona Durcik, Lenka Slav´ ıkov´ a, and Christoph Thiele. Local bounds for singular Brascamp- Lieb forms with cubical structure.Math. Z., 302(4):2375–2405, 2022.doi:10.1007/ s00209-022-03148-8

    work page 2022

  12. [12]

    Singular Brascamp-Lieb inequalities with cubical structure

    Polona Durcik and Christoph Thiele. Singular Brascamp-Lieb inequalities with cubical structure. Bull. Lond. Math. Soc., 52(2):283–298, 2020.doi:10.1112/blms.12310

    work page doi:10.1112/blms.12310 2020

  13. [13]

    Large values of the Gowers-Host-Kra seminorms.J

    Tanja Eisner and Terence Tao. Large values of the Gowers-Host-Kra seminorms.J. Anal. Math., 117:133–186, 2012.doi:10.1007/s11854-012-0018-2

    work page doi:10.1007/s11854-012-0018-2 2012

  14. [14]

    Some combinatorial, geometric and set theoretic problems in measure theory

    Paul Erd˝ os. Some combinatorial, geometric and set theoretic problems in measure theory. In D. K¨ olzow and D. Maharam-Stone, editors,Measure Theory, Oberwolfach 1983: Proceedings of the Conference held at Oberwolfach, June 26–July 2, 1983, volume 1089 ofLecture Notes in Mathematics, pages 321–327. Springer, Berlin, Heidelberg, 1984.doi:10.1007/BFb0072626

    work page doi:10.1007/bfb0072626 1983

  15. [15]

    Falconer and John M

    Kenneth J. Falconer and John M. Marstrand. Plane sets with positive density at infinity contain all large distances.Bull. London Math. Soc., 18(5):471–474, 1986.doi:10.1112/blms/18.5.471. 26

    work page doi:10.1112/blms/18.5.471 1986

  16. [16]

    The VC-dimension and point configurations inF 2 q.Discrete Comput

    David Fitzpatrick, Alex Iosevich, Brian McDonald, and Emmett Wyman. The VC-dimension and point configurations inF 2 q.Discrete Comput. Geom., 71(4):1167–1177, 2024.doi:10.1007/ s00454-023-00570-5

    work page 2024

  17. [17]

    Ergodic theory and configurations in sets of positive density

    Hillel Furstenberg, Yitzchak Katznelson, and Benjamin Weiss. Ergodic theory and configurations in sets of positive density. InMathematics of Ramsey theory, volume 5 ofAlgorithms Combin., pages 184–198. Springer, Berlin, 1990.doi:10.1007/978-3-642-72905-8\_13

    work page doi:10.1007/978-3-642-72905-8 1990

  18. [18]

    A new proof of Szemer´ edi’s theorem.Geom

    William Timothy Gowers. A new proof of Szemer´ edi’s theorem.Geom. Funct. Anal., 11(3):465– 588, 2001.doi:10.1007/s00039-001-0332-9

    work page doi:10.1007/s00039-001-0332-9 2001

  19. [19]

    Recent trends in Euclidean Ramsey theory.Discrete Math., 136(1-3):119– 127, 1994

    Ronald Lewis Graham. Recent trends in Euclidean Ramsey theory.Discrete Math., 136(1-3):119– 127, 1994. Trends in discrete mathematics.doi:10.1016/0012-365X(94)00110-5

    work page doi:10.1016/0012-365x(94)00110-5 1994

  20. [20]

    Nonconventional ergodic averages and nilmanifolds.Ann

    Bernard Host and Bryna Kra. Nonconventional ergodic averages and nilmanifolds.Ann. of Math. (2), 161(1):397–488, 2005.doi:10.4007/annals.2005.161.397

    work page doi:10.4007/annals.2005.161.397 2005

  21. [21]

    Iosevich, Gail Jardine, and Brian McDonald

    Alex. Iosevich, Gail Jardine, and Brian McDonald. Cycles of arbitrary length in distance graphs onF d q.Tr. Mat. Inst. Steklova, 314:31–48, 2021. English version published in Proc. Steklov Inst. Math.314(2021), no. 1, 27–43.doi:10.4213/tm4189

    work page doi:10.4213/tm4189 2021

  22. [22]

    The VC-dimension and point configurations inR d.arXiv preprint, 2025

    Alex Iosevich, Akos Magyar, Alex McDonald, and Brian McDonald. The VC-dimension and point configurations inR d.arXiv preprint, 2025. URL:https://arxiv.org/abs/2510.13984, arXiv:arXiv:2510.13984

    work page arXiv 2025

  23. [23]

    Chudnovsky and S

    Alex Iosevich, Brian McDonald, and Maxwell Sun. Dot products inF 3 q and the Vapnik- Chervonenkis dimension.Discrete Math., 346(1):Paper No. 113096, 9, 2023.doi:10.1016/j. disc.2022.113096

    work page doi:10.1016/j 2023

  24. [24]

    Boundedness of the twisted paraproduct.Rev

    Vjekoslav Kovaˇ c. Boundedness of the twisted paraproduct.Rev. Mat. Iberoam., 28(4):1143–1164, 2012.doi:10.4171/RMI/707

    work page doi:10.4171/rmi/707 2012

  25. [25]

    Density theorems for anisotropic point configurations.Canad

    Vjekoslav Kovaˇ c. Density theorems for anisotropic point configurations.Canad. J. Math., 74(5):1244–1276, 2022.doi:10.4153/S0008414X21000225

    work page doi:10.4153/s0008414x21000225 2022

  26. [26]

    Large dilates of hypercube graphs in the plane.Anal

    Vjekoslav Kovaˇ c and Bruno Predojevi´ c. Large dilates of hypercube graphs in the plane.Anal. Math., 50(3):893–915, 2024.doi:10.1007/s10476-024-00045-6

    work page doi:10.1007/s10476-024-00045-6 2024

  27. [27]

    Product of simplices and sets of positive upper density inR d.Math

    Neil Lyall and ´Akos Magyar. Product of simplices and sets of positive upper density inR d.Math. Proc. Cambridge Philos. Soc., 165(1):25–51, 2018.doi:10.1017/S0305004117000184

    work page doi:10.1017/s0305004117000184 2018

  28. [28]

    Distance graphs and sets of positive upper density inR d.Anal

    Neil Lyall and ´Akos Magyar. Distance graphs and sets of positive upper density inR d.Anal. PDE, 13(3):685–700, 2020.doi:10.2140/apde.2020.13.685

    work page doi:10.2140/apde.2020.13.685 2020

  29. [29]

    Weak hypergraph regularity and applications to geometric Ramsey theory.Trans

    Neil Lyall and ´Akos Magyar. Weak hypergraph regularity and applications to geometric Ramsey theory.Trans. Amer. Math. Soc. Ser. B, 9:160–207, 2022.doi:10.1090/btran/61

    work page doi:10.1090/btran/61 2022

  30. [30]

    ProQuest LLC, Ann Arbor, MI, 2023

    Brian McDonald.Configuration Problems Over Finite Fields and the Vapnik-Chervonenkis Dimension. ProQuest LLC, Ann Arbor, MI, 2023. Thesis (Ph.D.)–University of Rochester. URL:http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&rft_val_fmt=info: ofi/fmt:kev:mtx:dissertation&res_dat=xri:pqm&rft_dat=xri:pqdiss:30529399

    work page 2023

  31. [31]

    Brian McDonald, Anurag Sahay, and Emmett L. Wyman. The VC dimension of quadratic residues in finite fields.Discrete Math., 348(1):Paper No. 114192, 13, 2025.doi:10.1016/j.disc.2024. 114192. 27

    work page doi:10.1016/j.disc.2024 2025

  32. [32]

    Stein.Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 ofPrinceton Mathematical Series

    Elias M. Stein.Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, volume 43 ofPrinceton Mathematical Series. Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III

    work page 1993

  33. [33]

    Exploring the toolkit of Jean Bourgain.Bulletin of the American Mathematical Society, 58(2):155–171, 2021

    Terence Tao. Exploring the toolkit of Jean Bourgain.Bulletin of the American Mathematical Society, 58(2):155–171, 2021. Published electronically January 27, 2021.doi:10.1090/bull/1716

    work page doi:10.1090/bull/1716 2021

  34. [34]

    The uniform convergence of frequencies of the appearance of events to their probabilities.Teor

    Vladimir Naumovich Vapnik and Alexey Yakovlevich Chervonenkis. The uniform convergence of frequencies of the appearance of events to their probabilities.Teor. Verojatnost. i Primenen., 16:264–279, 1971. 28

    work page 1971