# Questions tagged [hausdorff-measure]

If $(X,\rho)$ is a metric space, then for any subset $S$, we have $$H_\delta^d(S):=\inf\ \left\{ \sum_{i=1}^\infty ({\rm diam}\ U_i)^d | S\subset \bigcup_{i=1}^\infty U_i,\ {\rm diam}\ U_i < \delta \right\},$$ where the infimum is over all countable covers of $S$. Then we have $H^d(S):=\lim_{\delta \rightarrow 0 } H^d_\delta (S)$, which is called the $d$-dimensional Hausdorff measure.

298 questions
Filter by
Sorted by
Tagged with
23 views

• 616
1 vote
38 views

61 views

### A question on the Hausdorff dimension of a subset of $\mathbb{R}.$ [closed]

Let $p\in [0,1].$ I am interested in showing that there exist sets $A,B\subset \mathbb{R}$ of Hausdorff dimension $p$ such that the $p$-dimensional Hausdorff measures $H_p(A)=\infty$ and $H_p(B)=0$. I'...
• 520
1 vote
104 views

### Local isometry and Hausdorff dimension

I am currently reading Falconer's book on Hausdorff dimension. My question is whether Hausdorff dimension is invariant under local isometry between smooth Riemannian manifolds? I think it should be ...
1 vote
35 views

### Embedded subshift in $[0,1]$ has dimension $0$?

I was wondering whether most embeddings of one-dimensional subshifts have zero Hausdorff dimension? Given a finite alphabet $\mathcal{A}= \{ 0,...,d-1 \}$ and $\Omega\subseteq \mathcal{A}^\mathbb{N}$, ...
• 7,805
1 vote
61 views

### Hausdorff dimension calculation for groups

I was reading about homogeneous space i.e let $G$ be a Lie group and $\Gamma$ be a discrete subgroup of $G$ with finite covolume. Then Hausdroff dimension of $G / \Gamma$ is the Hausdorff dimension of ...
• 170
1 vote
27 views

### what is the dimension of the intersection of the planar brownian range with a line

We consider two dimensional standard brownian motion $B: t\mapsto (B_{1}(t),B_{2}(t))$. Let $D$ be its range (that is the image of $[0,+\infty[$ by $B$ i.e, $B([0,+\infty[)$). Is there some known ...
• 11
23 views

### question related to hausdorff dimension and hausdorff measure [closed]

If $A$ is a subset of $\mathbb{R}^d$, we denote the diameter of $A$ by $\Delta(A)=\sup \{|x-y|: x, y \in A\} \in[0, \infty]$. For every $\alpha>0$ and every $\varepsilon>0$, we set, for every ...
• 103
83 views

### Naturalness of definition of line integral

Let $I = [a,b]$ denote some interval, and $f : I \rightarrow \mathbb{C}$ a continuous function of bounded variation, in other words, $f$ is a parameterisation of $\gamma = f(I)$, a rectifiable curve. ...
• 1,642
105 views

### Do there exist continuous maps from the Sphere to the Plane which preserve the Hausdorff Measure of all sets with some Dimension between 1 and 2?

It's well known that there do exist area preserving maps between the sphere and the plane. It's well known there do NOT exist distance preserving maps between the sphere and the plane. So naturally ...
• 17.5k
40 views

### References on Hausdorff Dimension on Non-Euclidean Vector Spaces?

Given a fractal set in the Euclidean space $\mathbb{R}^n$, we can study it's Hausdorff dimension. Common examples include: (1) The middle-thirds Cantor set in $\mathbb{R}$. (2) The Sierpinski Gasket ...
• 353
47 views

### Existence of an $\alpha$-regular measure with positive measure on a binary digits do not have a limiting frequency

let $$X=\left\{ \sum_{n=1}^{\infty}a_{n}2^{-n}:a_{n}\in\left\{ 0,1\right\} ,\liminf\frac{1}{n}\sum_{i=1}^{n}a_{i}<\limsup\frac{1}{n}\sum_{i=1}^{n}a_{i}\right\}$$ I'm studying fractal geometry and ...
1 vote