All Questions
Tagged with fractals hausdorff-dimension
31 questions
4
votes
0
answers
76
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An algebraic condition possibly related with the Hausdorff measure on $\mathbb{R}$
This is my first time to ask a question here. Please tell me if I can improve it.
I would like to introduce the following definition inspired from a measure theory exercise.
Definition. A subset $K$ ...
3
votes
1
answer
99
views
Intersection of IID fractal sets
Let $A, B \subset \mathbb R$ be IID random closed subset. Suppose that there exists $d \in (1/2, 1]$ such that the Hausdorff dimension of $A$ is equal to $d$ almost surely. Is it true that $\mathbf P\...
- 177
4
votes
1
answer
206
views
Existence of an $\alpha$-Hölder continuous function whose graph has positive Hausdorff measure of maximal dimension
It is standard that if $f:[0,1] \rightarrow \mathbb{R}$ is $\alpha$-Hölder continuous, then its graph has Hausdorff dimension at most $2-\alpha$. My naive expectation was that "most" graphs ...
- 45
3
votes
2
answers
142
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Dimension of the graph of a function $\varphi : \mathbb R^2 \to \mathbb R$
Let $\varphi : \mathbb{R}^2 \to \mathbb{R}$ be a continuous function, and let $G(\varphi)$ be the graph of $\varphi$. Denote $R:=\{(x,0) \in \mathbb{R}^2 | x \in \mathbb{R}\}$ as the real line in $\...
- 661
0
votes
1
answer
115
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Fractal sets and dimensions
Can we construct two sets $E$ and $F$ meeting the following criteria
$\dim_H(E) = \dim_H(F) = \dim_H(E ∩ F)$
$\dim_P(E), \dim_P(F)$, and $\dim_P(E ∩ F)$ are distinct?
Here $\dim_H$ denotes the ...
- 39
3
votes
1
answer
105
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 ...
7
votes
1
answer
173
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Plane curve with continuously increasing Hausdorff dimension
In a recent paper, we required the following fact.
Proposition 1. There exists a simple closed curve $\gamma\subset\mathbb{C}$ with the following property. If $\phi$ is a biholomorphic map, defined on ...
- 6,548
4
votes
0
answers
95
views
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 ...
- 20.7k
1
vote
1
answer
161
views
Is there a two-dimensional unimodal function with fractal level sets
Is there an open simply connected $U\subset\mathbb{R}^2$ and a continuous non-constant function $f: U\to \mathbb{R}$,
such that for all $c\in \mathbb{R}$ both sets
$$ f_{<c}~=~ f^{-1}\left( (-\...
- 1,676
5
votes
0
answers
160
views
Naïve definition of a measure on a fractal
This question was previously posted on MSE.
Let $K\subset \mathbb R^2$ be a compact fractal of Hausdorff dimension $1<d<2$. I want to define a natural measure on $K$.
One option would be to use ...
- 333
6
votes
1
answer
471
views
Does finite Hausdorff dimension imply finite packing dimension?
In other words, does there exist a metric space $(E,\rho)$ with finite Hausdorff dimension but infinite packing dimension?
Here are my thoughts:
I know that it is generally hard to relate Hausdorff ...
- 223
6
votes
1
answer
574
views
Direct proof that the set of badly approximable numbers have full Hausdorff dimension without using Schmidt games
A badly approximable number is an $x$ for which there is a positive constant $c$ such that for all rational $p/q$ we have
$$\left|{ x - \frac{p}{q} }\right| > \frac{c}{q^2} \ . $$
The set of badly ...
- 1,565
4
votes
2
answers
1k
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, ...
- 10.1k
2
votes
0
answers
99
views
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 ...
- 9,340
8
votes
1
answer
213
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How do sets with unit fractional Hausdorff measure of dimension $>1$ look like?
Triggered by the recent question How can we not know the measure of the Sierpiński triangle? I would like to ask:
Let $s>1$ and $s$ not be an integer. How to construct a set $A$ with $\mathfrak{H}^...
- 12.7k
2
votes
1
answer
216
views
Does fractallity depend on the Riemannian metric?
Edit: According to comment of Andre Henriques we revise the question:
In this question a fractal is a metric space whose topological and Hausdorff dimensions are different. So we would like that ...
- 366
11
votes
0
answers
337
views
Hausdorff dimension and von Neumann dimension
There are two subjects in which non-integral dimensions appear:
fractal geometry: consider the well-known Hausdorff dimension of fractals.
von Neumann algebra: consider a type ${\rm II_1}$ ...
2
votes
1
answer
254
views
Failure of Falconer distance problem in one dimension
I am told that the Falconer distance conjecture fails trivially in one dimension, but I really cannot find any reference for that. Precisely, I am asking the following question:
For a compact set $E\...
- 751
0
votes
0
answers
151
views
Set with modified lower box counting dimension strictly less than Hausdorff dimension
Please, can someone give (as simple as possible) example of the set for which modified lower box counting dimension is strictly smaller than Hausdorff dimension?
- 1
2
votes
0
answers
85
views
Intersections of Sierpinski carpets with lines
Let $S$ be the Sierpinski carpet contained in the square $[0,1]^2$. For Lebesgue almost every $a\in [0,1]^2$ and every $\theta\in\mathbb Q$ the intersection of the line $L_{a,\theta}$ with equation $y-...
- 743
7
votes
1
answer
209
views
Examples of probability measures with `fake' decay
To be concise, I am wondering whether there are natural examples of probability measures $\mu$ compactly supported on the real line which satisfy $\mu(I) \lesssim l_n^\alpha$ for all intervals $I$ ...
- 455
7
votes
1
answer
1k
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Hausdorff dimension of the graph of an increasing function
Let $f$ be a continuous, strictly increasing function from $[0,1]$ to itself with $f(0)=0, f(1)=1$. Let $\Gamma_f$ denote its graph. What can be said about the Hausdorff dimension of $\Gamma_f$? In ...
- 2,135
1
vote
0
answers
114
views
density of fractal measures
Let $s\in (0, 1)$ be a real number. Let $E\subset [0, 1]$ be a Borel set whose Hausdorff dimension is given by $s$. Assume that $\mathcal{H}^s(E)=+\infty$, that is, the $s$-dimensional Hausdorff ...
- 11
0
votes
0
answers
122
views
Hausdorff dimension of $X\times X$
I am thinking of the following question:
Let $X\subseteq \mathbb R$. Is it true that
$$
\mathrm{dim_H}(X\times X)=2\mathrm{dim_H}(X)?
$$
My thoughts:
We know that $\mathrm{dim_H}(X)+\mathrm{dim_H}(...
- 751
6
votes
5
answers
1k
views
Fractals of dimension zero
Are there any famous examples of fractals, or other closed sets, of cardinality continuum but Hausdorff dimension 0?
I can think of something ad hoc like a Cantor middle $\frac13$ set where the ...
- 24.8k
5
votes
1
answer
511
views
Hausdorff dimension of boundaries of open sets diffeomorphic to $\mathbb{R}^n$
Let $B$ be a bounded open subset of $\mathbb{R}^n$ which is diffeomorphic to $\mathbb{R}^n$. (I am not sure how important the diffeomorphism is but this is the case I am interested in.) Let $C$ be its ...
- 1,167
4
votes
1
answer
969
views
Usable Change-of-Variables Formula for Hausdorff Measure
Let $H^{s}$ be the $s$-dimensional Hausdorff measure, let $D$ be a nonsingular matrix. Consider the change of measure formula:
$$
\int\limits_{A} f(Dx) \; \mathrm{d}H^{s}(x) = \int\limits_{ D A} f(y)...
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 $F\subset \mathbb{R}^m$
$$
\dim_HF+\dim_HE\leq \dim_H(E\times F)\leq \dim_HE+\...
user11178
8
votes
2
answers
388
views
Isometrically-invariant measures and dilation of the Cantor set
Let $C$ be the Cantor middle-thirds set. Let $\mu$ be a finitely-additive isometrically-invariant measure on all subsets of $\mathbb R$. Then $\mu(3C)=2\mu(C)$, where $aB = \{ ax : x \in B \}$. ...
- 2,555
2
votes
1
answer
856
views
Hausdorff dimension of a subset of Cantor set
What is the Hausdorff dimension of the subset
$$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 ...
- 153
5
votes
5
answers
2k
views
Fractal questions: Weierstraß-Mandelbrot
Coming from a specific field in algebraic geometry, I am a total noob in Fractal Theory and I'd like to learn it a bit. I hope I am tolerated for my maybe-trivial questions. I just read about the ...
- 2,275