  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.
