  Generalising a construction of Falconer, we consider classes of
$G_\delta$-subsets of $\mathbb{R}^d$ with the property that sets belonging to
the class have large Hausdorff dimension and the class is closed under
countable intersections. We relate these classes to some inhomogeneous
potentials and energies, thereby providing some useful tools to determine if a
set belongs to one of the classes.
  As applications of this theory, we calculate, or at least estimate, the
Hausdorff dimension of randomly generated limsup-sets, and sets that appear in
the setting of shrinking targets in dynamical systems. For instance, we prove
that for $\alpha \geq 1$, \[ \mathrm{dim}_\mathrm{H}\, \{ \, y : | T_a^n (x) -
y| < n^{-\alpha} \text{ infinitely often} \, \} = \frac{1}{\alpha}, \] for
almost every $x \in [1-a,1]$, where $T_a$ is a quadratic map with $a$ in a set
of parameters described by Benedicks and Carleson.

Dynamical Systems;Classical Analysis and ODEs