Questions tagged [hausdorff-dimension]

Questions about dimensions of possibly highly irregular or "rough" sets, Hausdorff–Besicovitch dimension and related concepts such as box-counting or Minkowski–Bouligand dimension. To be used with [tag:fractals] or [tag:dimension-theory].

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How to define a notion of Hausdorff homeomorphism?

A separable metric space is called fractal if its Hausdorff and topological dimensions are different. The Hausdorff dimension is not invariant by homeomorphism (see this post). Question: How to ...
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Rational points in $\mathbb{R}^2$ on a set of lower Hausdorff dimension

Suppose $X \subseteq \mathbb{R}^2$ is a set of Haudorff dimension less than or equal to $1$. Suppose also that $X$ is compact. I was wondering can we obtain a bound for the following quantity? $$ \#\{ ...
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Does a curve with a box-counting dimension greater than 1 have to have infinite length?

If I have a curve that occupies a finite space (e.g. the unit square) and it has a box-counting dimension > 1, can it still have a finite length? If not, is there a proof of this?