We consider sets in uniformly perfect metric spaces which are null for every doubling measure of the space or which have positive measure for all doubling measures. These sets are called thin and fat, respectively. In our main results, we give sufficient conditions for certain cut-out sets being thin or fat.
We calculate the almost sure Hausdorff dimension of the random covering set in -dimensional torus , where the sets are parallelepipeds, or more generally, linear images of a set with nonempty interior, and are independent and uniformly distributed random points. The dimension formula, derived from the singular values of the linear mappings, holds provided that the sequences of the singular values are decreasing.
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