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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.

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### Counting fractals modulo "shared complements"

Previously asked at MSE: Let $\mathscr{H}$ be the space of compact nonempty subsets of $\mathbb{R}^2$ (I'm not especially wedded to dimension $2$, so feel free to tweak that if it would lead to a more ...
1 vote
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### Fubini's theorem for Hausdorff measures

$B\subset \mathbb{R}^2$ is a Borel set. Define the slices $B_x:= \{y \in \mathbb{R}: (x,y) \in B \}$. If $\lambda$ denotes the Lebesgue measure on $\mathbb{R}$, presentations of Fubini's theorem often ...
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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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### 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: ...
532 views

### 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 ...
966 views

### 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 ...
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 ...