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Given a non-empty bounded subset of hyperbolic space and a Kleinian groupacting on that space, the orbital set is the orbit of the given set under theaction of the group. We may view orbital sets as bounded (often fractal)subsets of Euclidean space. We prove that the upper box dimension of an orbitalset is given by the maximum of three quantities: the upper box dimension of thegiven set; the Poincar\'e exponent of the Kleinian group; and the upper boxdimension of the limit set of the Kleinian group. Since we do not make anyassumptions about the Kleinian group, none of the terms in the maximum can beremoved in general. We show by constructing an explicit example that the(hyperbolic) boundedness assumption on $C$ cannot be removed in general.

Dimensions of Kleinian orbital sets