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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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The Hausdorff dimension of the zero set of a real analytic function

Let $n>1$, and let $f:\mathbb{R}^n \to \mathbb{R}$ be a real-analytic function which is not identically zero. Does $\dim_{\mathcal H}(f^{-1}(0)) \le n-1$? here $\dim_{\mathcal H}$ refers to the ...
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Does the domain where a harmonic map is conformal has Hausdorff dimension smaller than one?

Let $\mathbb{D}^2$ be the closed unit disk, and let $f:\mathbb{D}^2 \to \mathbb{R}^2$ be a harmonic map, satisfying $\det df \neq 0$ almost everywhere on $\mathbb{D}^2$. Suppose also that $$U= \{ p \...
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Covering dimension of boundary of compact subset of $\mathbb{R}^n$

Let $X$ be a compact subset of $\mathbb{R}^n$, with the inherited Euclidean topology. Does it follow that $\dim_{cov}(\partial X)\leq n-1$? I would be happy to have a reference for that, if it is ...
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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?