All Questions
Tagged with hausdorff-dimension reference-request
13 questions
23
votes
3
answers
1k
views
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 ...
- 1,045
5
votes
1
answer
511
views
Hausdorff dimension of boundaries of open sets diffeomorphic to $\mathbb{R}^n$
Let $B$ be a bounded open subset of $\mathbb{R}^n$ which is diffeomorphic to $\mathbb{R}^n$. (I am not sure how important the diffeomorphism is but this is the case I am interested in.) Let $C$ be its ...
- 1,167
5
votes
0
answers
586
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 ...
- 779
4
votes
2
answers
279
views
Hausdorff dimension of sequence space
Let $\Omega =\{0,1\}^{\mathbb{N}}$ denote the set of infinite sequences with elements $0$ or $1$. Let $d$ be the metric on $\Omega$ given by $d((x_n),(y_n))=1/2^m$, where $m=\min\{i\in\mathbb{N}\,:\,...
- 83
4
votes
1
answer
245
views
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 ...
- 4,459
4
votes
1
answer
196
views
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 ...
- 687
4
votes
1
answer
903
views
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: ...
- 649
4
votes
0
answers
119
views
A quantity that distinguishes finer than Hausdorff dimension
Consider sets $A\subseteq \mathbb{R}$ with Lebesgue measure zero and Hausdorff dimension one. For instance the set of real numbers with bounded entries in their continued fraction expansion have ...
- 1,648
3
votes
0
answers
204
views
Product Fractals
Here is a theorem found in the Falconer's book on fractal geometry:
Theorem: For any sets $E\subset \mathbb{R}^n$ and $F\subset \mathbb{R}^m$
$$
\dim_HF+\dim_HE\leq \dim_H(E\times F)\leq \dim_HE+\...
user11178
2
votes
1
answer
388
views
Construction of null sets with prescribed Hausdorff dimension and generalizations
Given $h:\mathbb{R}_0^+ \to \mathbb{R}_0^+$ increasing and right continuous, the outer measure $\mathcal{H}^h$ in $\mathbb{R}^d$ that assigns to every $E\subset\mathbb{R}^d$ the measure $$\mathcal{H}^...
- 193
2
votes
1
answer
127
views
Hausdorff dimension and non-empty intersections with lines
Let $A\subseteq [0,1]^d$, $d\geq 2$, a set with Hausdorff dimension $\operatorname{dim}_{\mathcal{H}}A=s$. What is the minimum $s$ (if any) which guarantee that $A$ has non-empty intersections with a ...
- 282
1
vote
1
answer
296
views
Hausdorff dimension of the non-differentiability set of a locally Lipschitz function
Let $f:\mathbb R^n \to \mathbb R$ and $E := \{x \in X : f \text{ not Fréchet differentiable at }x\}$. Then $E$ is Borel measurable. It is well-known that
Theorem If $f$ is convex, then the Hausdorff ...
- 825
1
vote
1
answer
148
views
Dimension of quotient of compact totally disconnected group action
Assume that $X$ is a compact metric space and $G$ is compact
totally disconnected group. And $X$ has isometric free $G$-action
i.e. $gx=x\Rightarrow g=e$.
Then the following holds $${\rm dim}\ ...
- 1,100