Let us focus on the plane. I am wondering whether the sets with small (but non zero) Hausdorff dimension share some interesting properties.

Let $\epsilon\in(0,1)$ and assume that a set $F\subset \Bbb R^2$ is such that $0<\dim_{\mathcal H}(F)<\epsilon$. Then ...

This is a pretty vague question but I would like to know if there are interesting properties which hold in this setup, like an interesting covering property for example...


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