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

Inhomogeneous potentials, Hausdorff dimension and shrinking targets