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Joint upper Banach density, VC dimensions and Euclidean point configurations

Bruno Predojević

TL;DR

This work introduces joint density notions $\overline{\delta}(A,B)$ and $\overline{\delta}_{VC}(A,B)$ to study two coupled planar subsets and derives two main results: a two-set Szekély-type theorem guaranteeing large distances between $A$ and $B$ under positive joint density, and a VC-dimension result showing that translates of a convex, centrally symmetric curve $\Gamma$ intersected with $A$ exhibit maximal VC-dimension on large scales when $\overline{\delta}_{VC}(A,B)>0$. The proofs combine Fourier-analytic methods, Gaussian smoothing, Gowers uniformity norms, and singular Brascamp–Lieb inequalities, organizing key counting forms into structured, error, and uniform components and bounding each. These results connect density largeness to VC-dimension phenomena in Euclidean plane geometry, providing Ramsey-type conclusions for two-set distance configurations and flexible curve configurations.

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 $Γ$ which bounds a convex and compact region in the plane and is of non-vanishing curvature, the family consisting of portions of translates of $tΓ$ has the maximal possible Vapnik--Chervonenkis dimension.

Joint upper Banach density, VC dimensions and Euclidean point configurations

TL;DR

This work introduces joint density notions and to study two coupled planar subsets and derives two main results: a two-set Szekély-type theorem guaranteeing large distances between and under positive joint density, and a VC-dimension result showing that translates of a convex, centrally symmetric curve intersected with exhibit maximal VC-dimension on large scales when . The proofs combine Fourier-analytic methods, Gaussian smoothing, Gowers uniformity norms, and singular Brascamp–Lieb inequalities, organizing key counting forms into structured, error, and uniform components and bounding each. These results connect density largeness to VC-dimension phenomena in Euclidean plane geometry, providing Ramsey-type conclusions for two-set distance configurations and flexible curve configurations.

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 and for a smooth, closed, centrally symmetric, planar curve which bounds a convex and compact region in the plane and is of non-vanishing curvature, the family consisting of portions of translates of has the maximal possible Vapnik--Chervonenkis dimension.
Paper Structure (16 sections, 16 theorems, 134 equations, 2 figures)

This paper contains 16 sections, 16 theorems, 134 equations, 2 figures.

Table of Contents

  1. Introduction
  2. A brief overview of previous results
  3. Statement of the first main result
  4. VC dimension and statement of second main result
  5. Organization of the paper
  6. Preliminaries and notation
  7. Gaussian functions and Fourier analysis
  8. Geometry of curves in the plane
  9. Common generalization of $\overline{\delta}(A,B)$ and $\overline{\delta}_{VC}(A,B)$
  10. Proof of Theorem \ref{['thm: two set Szekely qualitative']}
  11. Proof of Theorem \ref{['thm: VC dimension, main theorem']}
  12. The Structured Part bound
  13. The Error Part bound
  14. A particular case of singular Brascamp--Lieb inequalities
  15. The Uniform Part bound
  16. ...and 1 more sections

Key Result

Theorem 1.2

Let $A,B \subseteq \mathbbm{R}^2$ be measurable sets such that $\overline{\delta}(A,B) > 0$. Then, there exists a large enough $\lambda_0=\lambda_0(A,B) > 0$ such that for all $\lambda \geqslant \lambda_0$, there exist $x \in A$ and $y \in B$ such that

Figures (2)

  • Figure 1: Flexible configuration from Theorem \ref{['thm: VC dimension, configuration variant qualitative']}
  • Figure 2: Sketch of proof of Lemma \ref{['lem: Critical convexity']}

Theorems & Definitions (32)

  • Definition 1.1
  • Theorem 1.2: Two-set Szekély's problem
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6
  • Theorem 1.7: VC dimension, configuration variant
  • Lemma 2.1: Essential disjointness of Gaussian functions
  • Lemma 2.2: Fourier dimension estimate
  • Proposition 2.3
  • ...and 22 more