Joint upper Banach density, VC dimensions and Euclidean point configurations

TL;DR

This paper extends the concept of upper Banach density to pairs of sets in the plane, linking it to large-distance realizations and VC dimension of scaled convex curves.

Contribution

It introduces two generalized density measures and connects them to geometric configurations and VC dimension bounds for scaled convex curves.

Findings

Generalized density measures enable large-distance results for two sets.

For large scales, the VC dimension of translated scaled convex curves is maximal.

The results unify density, geometric, and combinatorial properties in the plane.

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.