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Triggered by the recent question How can we not know the measure of the Sierpiński triangle? I would like to ask: Let $s>1$ and $s$ not be an integer. How to construct a set $A$ with $\mathfrak{H}^...

Are there any famous examples of fractals, or other closed sets, of cardinality continuum but Hausdorff dimension 0? I can think of something ad hoc like a Cantor middle $\frac13$ set where the ...

Here is a theorem found in the Falconer's book on fractal geometry: Theorem: For any sets $E\subset \mathbb{R}^n$ and $F\subset \mathbb{R}^m$ $$ \dim_HF+\dim_HE\leq \dim_H(E\times F)\leq \dim_HE+\...