# 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.

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### Hausdorff dimension and critical exponent of words

What is the Hausdorff dimension of the subset $S_c \subset [0,1]$ of points such that the critical exponent of their binary expansion is $c$? It's clear that $\dim_H S_{\infty}=1$, but what can be ...
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### Maximal Hausdorff dimension of the set on which derivatives do not agree

Let $f, g: [0, 1] \to \mathbb R$ be functions that are differentiable a.e. with $f’ = g’$ almost everywhere. What is the maximal Hausdorff dimension $d$ (and corresponding Hausdorff $d$-measure) of ...
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### Hausdorff dimension of Julia set

Can anyone show me the proof "Hausdorff dimension of Julia set is strictly positive"? For purpose to prove this we might have to prove the green function of basin of attraction to infinity ...
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### Investigating a 1/2 -dimensional sphere and defining a fractional Euclidean space

A small note I'm a new member on MO and I'm not sure if this question fits in here. If not, please don't be too hard on me. I am transferring a question I asked on MSE in here, because the user who ...
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### Hausdorff dimension and surface measure

Could someone please indicate me some reference that contains the proof of the following theorem? Below $\mathcal{H}^n$ denotes the $n$-dimensional Hausdorff outer measure in $\mathbb{R}^n$. Theorem: ...
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### On the Hausdorff dimension of a Cantor set

In what follows I refer to this paper by Orevkov. I am writing a paper on this, so if somebody is interested we could consider to write a joint paper. Consider a sequence $R=\{R_n\}_n$ of strictly ...
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### (1, 2) stability and Hausdorff dimension

Recall that a set $E \subset \mathbb{R}^n$ is called (1,2) stable if it satisfies the following: $$W_0^{1,2}(E) = W_0^{1,2}(E^0),$$ where $E^0$ is the interior of $E$ and $W_0^{1,2}(E)$ is defined ...
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### A set whose Hausdorff dimension gradually changes?

Can there be a set whose Hausdorff dimension gradually changes? For instance, a set of real numbers contained in an interval, whose Hausdorff dimension is 0 at the beginning and 1 closer to the end, ...
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### Hausdorff dimension between $(1,2)$

Is there a number $c \in (1,2)$ for which there exist some interesting geometric property/properties which hold for every set of Hausdorff dimension in $(1,c)$ and does not hold for any set of ...
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### Hausdorff dimension of the boundary of fibres of Lipschitz maps

Let $f: \mathbb{R}^m\rightarrow \mathbb{R}^{m-k}$ be a Lipschitz map. Can we get a uniform estimate on the Hausdorff dimension of the boundaries of fibres of $f$? I.e. do we have an upper bound for ...
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### Existence of subset with given Hausdorff dimension

Let $A\subseteq \mathbb{R}$ be Lebesgue-measurable and let $0<\alpha<1$ be its Hausdorff dimension. For a given $0<\beta <\alpha$ can we find a subset $B\subset A$ with Hausdorff ...
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### Is there a characterization of the Hausdorff measures?

It is known that there is a unique measure on the Borel $\sigma$-algebra of $\mathbb{R}^n$ such that the measure of the rectangle $\prod_i [a_i,b_i[$ is $\prod_i (b_i-a_i)$. This is the Lebesgue ...
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### volume entropy and Hausdoff dimension

In relation to this question: Relation between volume entropy and Hausdorff dim of limit set? Given a metric space $Z$ and a hyperbolic approximation $X := hyp_{r_0}(Z)$ (as defined for example here)....
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### Is there a concept of uniform Hausdorff dimension?

Let $M$ be a metric space and let $U \subset M$ be open. Then the Hausdorff dimension of $U$ is defined in the usual way. If there is a single dimension number $d$ that is the Hausdorff dimension of ...