Geometrical sets with forbidden configurations

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.