1
vote
1answer
24 views

Generalisation of Vitali's covering lemma

In "The geometry of fractal sets", Falconer gives the following generalisation of the Vitali covering lemma as an exercise: Let $\mu$ be any measure on $\mathbb{R}^{n}$ and $E$ a set with ...
6
votes
1answer
139 views

Definitions of Sierpinski Carpet and Higher Dimensional Analogues

We define the Cantor Set as: $Let \mathscr{J} := \{ 0, 2, \ldots , 3^{m-1} -1 \}$ for $m \in \mathbb{N}$, then $$C = [0,1] \setminus \bigcup_{m \in \mathbb{N}} \bigcup_{k \in \mathscr{J}} \Big( ...
0
votes
0answers
53 views

Cantor sets, Sierpinski carpets, and Menger sponge

How does one distiguish between iterations near infinity? Naturally there is an empty feeling about saying that the iterations $\forall k \in \mathbb{N} < \infty$, $\infty + k$ and $\infty - k$ are ...
1
vote
1answer
116 views

What is the area of the apollonian gaskets?

I searched for the internet, but found nothing relavant to the area. The areas in each intermediate step form a bounded increasing sequence, so there is a limit. But wil it eventually fill in almost ...
0
votes
1answer
142 views

Hausdorff Measure and Hausdorff Dimension

Could someone explain the intuition behund the Hausdorff Measure and Hausdorff Dimension? The Hausdorff Measure is defined as the following: Let $(X,d)$ be a metric space. $\forall S \subset X$, ...
7
votes
0answers
194 views

About devil's staircases

We say that a function $f:\left[a,b\right] \to \mathbb{R}$ is a singular function or a devil's staircase if $f$ satisfies the following properties: $f$ is continuous; $f(a) < f(b)$; $f$ is ...
8
votes
1answer
328 views

Regular open set whose boundary has nonzero volume.

I found this question quite interesting, but its answers were disappointingly non-geometric. I'd be interested to know whether there exists a geometric example. To be precise about what I mean by a ...
4
votes
0answers
67 views

Is the measure induced by the Mandelbrot set computable on rational rectangles?

Is there a computable function that, given a positive rational number $\epsilon$ and a rectangle with rational corners $A$ returns a number $f(A,\epsilon)$ such that $|\mu(A \cap ...
0
votes
1answer
215 views

Relation between Hausdorff metric of and Hausdorff measure of non-compactness

Let $(X,d)$ be a metric space and let $$K(X)=\lbrace Y\subset X\colon Y\text{ is non-empty and compact}\rbrace.$$ Endow $K$ with the Hausdorff metric (which is the natural metric on this space, see ) ...