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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4
votes
1answer
71 views

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 \...
3
votes
3answers
64 views

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 ...
3
votes
1answer
37 views

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 ...
3
votes
1answer
23 views

Hausdorff dimension of compact set

Let $(X, d)$ be a metric space and $S \subset X$. Let $\DeclareMathOperator{\diam}{diam}\diam(S)$ denote diameter of $S$, that is $\diam(S) = \sup \{ d(x, y) \colon \: x, y \in S \}$. Let $\delta > ...
1
vote
1answer
30 views

Similarity dimension and Hausdorff dimension

How do we prove the similarity dimension equals the Hausdorff dimension if the self-similar set satisfies the open set condition? Which article contains this proof?
1
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0answers
47 views

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 ...
0
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1answer
33 views

Hausdorff dimension of radial set

Let $A\subset [0,1]$ be a compact set of Hausdorff dimension $\alpha$. Let also $A_n:=\{x\in \mathbb{R}^n~:~ \|x\|_2 \in A\}$. Is it true that $\text{dim}_H(A_n)=n-1+\alpha$? I believe that this ...
0
votes
0answers
14 views

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? $$ \#\{ ...
0
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0answers
19 views

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?