The hausdorff-dimension tag has no usage guidance.

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### Hausdorff dimension of a Cantor-like set

Suppose $K$ is a subset of $[0,1]$ with the following property: for almost $x,y \in K$, we have
$$\frac{x+y}{2} \not\in K.$$
(Here, "almost in $K$" means "in $K$ except for a countable subset").
...

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70 views

### Distortion of the Hausdorff dimension of sums of Cantor sets under local scaling

The following question deals with possible distortion of the Hausdorff dimension of sums of Cantor sets as one "zooms in" on the sum around any given point.
Let us assume that $C_1$ and $C_2$ are two ...

**8**

votes

**1**answer

162 views

### Best Hölder exponents of surjective maps from the unit square to the unit cube

The Peano's square-filling curve $p:I\to I^2$ turn's out to be Hölder continuous with exponent $1/2$ on the unit interval $I$ (a quick way to see it, is to note that $p$ is a fixed point of a ...

**5**

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**1**answer

165 views

### Calculate Hausdorff measure with Frostman measures

Fix a metrix space $(X,d)$ and consider the Hausdorff (outer) measures $\mathcal{H}^s$ on $X$.
A Frostman measure on $X$ is a finite Borel measure $\mu$ such that there exists $C,t,r_0>0$ with ...

**3**

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**1**answer

250 views

### The relation between Hausdorff dimension of an $n$-manifold and $n$

It is known that for a topological space with different metrics, the Hausdorff dimensions may not be equal in general.
For the case of manifolds, suppose $M$ is a $n$-manifold with a ...

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451 views

### Hausdorff dimension of R x X

In general, the Hausdorff dimension of a product is at least the sum of the dimensions of the two spaces. Does equality hold if one space is Euclidian?
So let $X$ be a metric space and let ...

**6**

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**1**answer

358 views

### Geometric measures different from Hausdorff

$\newcommand{\RR}{\mathbb{R}}\newcommand{\calF}{\mathcal{F}}\newcommand{\diam}{\mathrm{diam}}$
In geometric measure theory there are various notions of $m$-dimensional measure for sets $A\subset ...

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257 views

### Are there additive subgroups of reals of dimension 1 with no subgroups of dimension strictly between 0 and 1?

I will use $dimA$ to denote the Hausdorff dimension of a set $A \subseteq \mathbb{R}$. Being a null set means having Lebesgue measure zero.
In the 1966 paper "Additive gruppen mit vorgegebener ...

**5**

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284 views

### Can Hausdorff dimension make sets into a Tropical Semiring?

If $X$ is a metric space, we construct Hausdorff $d$ measure from the outer measure
\begin{equation}
H^d(U) = \lim_{\delta \to 0}\inf\left\{\sum_{i=1}^\infty \left(\text{diam}(E_i)\right)^d : ...

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**2**answers

110 views

### Hausdorff measure of the zero set

Let $f : \mathbb R^n\to \mathbb R$ continuous, for which there exist $x,y\in\mathbb R^n$, such that $f(x)f(y)<0$.
Is it true that the Hausdorff dimension of the zero set of $f$ is at least $n-1$?

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**1**answer

193 views

### Are there any good techniques for calculating Hausdorff measure?

I'm aware that many techniques have been developed for the purpose of calculating Hausdorff dimension (although I'm fairly unfamiliar with them), but my question is whether or not we have any good ...

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**1**answer

237 views

### Hausdorff metric on C[0,1]

Let us consider $C[0,1]$, the space of continuous functions $f\colon [0,1] \to \mathbb{R}$. It comes usually with the metric of the maximum, or of the supremum, $d_{L^{\infty}}$. Each element $f$ in ...

**5**

votes

**1**answer

124 views

### Multifractal Analysis and Dimension Spectrum of Unions

Consider the classical Multifractal Analysis, and the decomposition of the state space $X$ into level sets
$$X=\bigcup_{\alpha}\left\{x\mid d_\mu(x)=\alpha\right\}\cup\left\{x\mid d_\mu(x) ...

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192 views

### Isometrically-invariant measures and dilation of the Cantor set

Let $C$ be the Cantor middle-thirds set. Let $\mu$ be a finitely-additive isometrically-invariant measure on all subsets of $\mathbb R$. Then $\mu(3C)=2\mu(C)$, where $aB = \{ ax : x \in B \}$. ...

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319 views

### Hausdorff dimension of Julia sets of quadratics not in the Mandelbrot set.

What are the bounds on the possible values of the Hausdorff dimension of the Julia sets of quadratics not in the Mandelbrot set? In particular, assume we have a quadratic $q_c: z \mapsto z^2 + c$ on ...

**2**

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**1**answer

454 views

### Hausdorff dimension of a subset of Cantor set

What is the Hausdorff dimension of the subset
$$F := \{ x = \sum^\infty_{n=1} \frac{2 x_n}{3^n} \in [0,1] : x_n \in \{ 0 , 1 \} , x_n = 1 \Rightarrow x_{n+1}=0 \}$$
of the Cantor set? Is it known ...

**5**

votes

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420 views

### Quantitative measurement of infinite dimensionality

I recently encountered the metric mean dimension, which is a numerical metric invariant of (discrete time, compact space) dynamical systems that refines topological entropy for infinite-entropy ...

**2**

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626 views

### Simple definition of the Hausdorff measure using squared paper

I am giving a "non-technical" seminar in which I would like to give an elementary introduction to the Hausdorff dimension and measure.
For simplicity, I was hoping to give a more intuitive ...