# Are there interesting general properties held by sets with small Hausdorff dimension?

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...