  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 $P_i$. In this paper we will study 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, $\mathbf{m}_{\mathbb{R}^d}(t_1 P_1, t_2 P_2, \dots,
t_n P_n)$ tends to $\prod_{i=1}^n \mathbf{m}_{\mathbb{R}^d}(P_i)$ as the ratios
between consecutive dilation parameters $t_{i+1}/t_i$ 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.
