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Graduate Range of the Hausdorff dimension
in \mathbb{R}^n, you will not find any set that has a dimension >n ! To construct a set with Hausdorf dimension 0 < d < n, it suffices to construct a set E that has a d-Hausdorf measure 0 < \mathcal{H}^{(d)}(E) < \infty. If you know the example of Cantor-like sets in \mathbb{R}, you can adapt...