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Joint upper Banach density, VC dimensions and Euclidean point configurations

T0 review · 0 major / 3 minor · reviewed 2026-05-18 · grok-4.3

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Pith's one-line read 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.

desk verdict The paper defines two joint upper Banach densities for plane sets and uses one to show that portions of large scaled translates of a smooth convex curve attain maximal VC dimension. read the letter →

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

Bruno Predojević This is my paper · ORCID
classification math.CAmath.CO
keywords jointupperBanachdensityVCdimensionconvexcurvespointconfigurationsEuclideandistancesmeasurablesetsVapnik-Chervonenkis
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 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.

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.

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.

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Extended reading notes

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.

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.

Editorial extensions

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.

Reading between the lines

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.
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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

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 · score 0.0 of 10

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.

Assumptions & free parameters 0 free parameters · 1 assumptions · 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.

assumptions (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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Cite this review

Pith. "Pith review of Joint upper Banach density, VC dimensions and Euclidean point configurations." pith.science (2026). https://pith.science/paper/2510.17453

@misc{pith2026251017453,
  author       = {Pith},
  title        = {Pith review of: Joint upper Banach density, VC dimensions and Euclidean point configurations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2510.17453}},
  note         = {Machine review of arXiv:2510.17453}
}
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.

Figures

Figures reproduced from arXiv: 2510.17453 by the authors.

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 positive up… view at source ↗
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 ↗

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