Geometrical sets with forbidden configurations

TL;DR

This paper introduces the concept of independence density for geometric configurations in Euclidean space and on spheres, analyzing how it behaves under dilation and generalizing previous theorems in geometric Ramsey theory.

Contribution

It defines the independence density for geometric configurations and studies its asymptotic behavior under dilation, extending results to spherical settings.

Findings

Independence density tends to the product of individual densities as dilation ratios increase.

Dilation ratios cause an exponential decay in the maximum density of sets avoiding configurations.

Results generalize theorems by Bourgain and Bukh in geometric Ramsey theory.

Abstract

Given finite configurations $P_1, \dots, P_n \subset \mathbb{R}^d$, let us denote by $\mathbf{m}_{\mathbb{R}^d}(P_1, \dots, P_n)$ the maximum density a set $A \subseteq \mathbb{R}^d$ can have without containing congruent copies of any $P_i$. We will initiate the study of this geometrical parameter, called the independence density of the considered configurations, and give several results we believe are interesting. For instance we show that, under suitable size and non-degeneracy conditions, $\mathbf{m}_{\mathbb{R}^d}(t_1 P_1, t_2 P_2, \dots, t_n P_n)$ progressively `untangles' and tends to $\prod_{i=1}^n \mathbf{m}_{\mathbb{R}^d}(P_i)$ as the ratios $t_{i+1}/t_i$ between consecutive dilation parameters grow large; this shows an exponential decay on the density when forbidding multiple dilates of a given configuration, and gives a common generalization of theorems by Bourgain and by Bukh in geometric Ramsey theory. We also consider the analogous parameter $\mathbf{m}_{S^d}(P_1, \dots, P_n)$ on the more complicated framework of sets on the unit sphere $S^d$, obtaining the corresponding results in this setting.