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Inhomogeneous potentials, Hausdorff dimension and shrinking targets

Generalising a construction of Falconer, we consider classes of 𝐺𝛿-subsets of ℝ𝑑 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