Cartesian products of the Sierpiński carpet (and similar self-similar fractals) with itself at least twice do not attain their conformal dimension.
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Sets with d-dimensional weak tangent fields are mapped by typical 1-Lipschitz maps to Euclidean space into sets of Hausdorff dimension at most d, up to measure zero.
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Cartesian products of Sierpi\'nski carpets do not attain their conformal dimension
Cartesian products of the Sierpiński carpet (and similar self-similar fractals) with itself at least twice do not attain their conformal dimension.
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Perturbations of measures and sets having curves in d directions
Sets with d-dimensional weak tangent fields are mapped by typical 1-Lipschitz maps to Euclidean space into sets of Hausdorff dimension at most d, up to measure zero.