All Questions
Tagged with hausdorff-dimension fractals
31 questions
4
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2
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212
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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 ...
- 565
4
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0
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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
105
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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 ...
CommunityBot
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3
votes
1
answer
99
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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\...
- 23.2k
3
votes
2
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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 $\...
- 2,093
5
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5
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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 ...
- 91.8k
0
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1
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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 ...
8
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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}^...
- 28k
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 ...
- 91.8k
4
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0
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95
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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
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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
$$ f_{<c}~=~ f^{-1}\left( (-\...
- 1,676
5
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0
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160
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Naïve definition of a measure on a fractal
This question was previously posted on MSE.
Let $K\subset \mathbb R^2$ be a compact fractal of Hausdorff dimension $1<d<2$. I want to define a natural measure on $K$.
One option would be to use ...
- 333
6
votes
1
answer
471
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Does finite Hausdorff dimension imply finite packing dimension?
In other words, does there exist a metric space $(E,\rho)$ with finite Hausdorff dimension but infinite packing dimension?
Here are my thoughts:
I know that it is generally hard to relate Hausdorff ...
- 41.1k
6
votes
1
answer
574
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Direct proof that the set of badly approximable numbers have full Hausdorff dimension without using Schmidt games
A badly approximable number is an $x$ for which there is a positive constant $c$ such that for all rational $p/q$ we have
$$\left|{ x - \frac{p}{q} }\right| > \frac{c}{q^2} \ . $$
The set of badly ...
- 8,842
4
votes
2
answers
1k
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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, ...
- 1,329
2
votes
0
answers
99
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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
11
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0
answers
337
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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
216
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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 ...
- 28k
7
votes
1
answer
209
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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
2
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1
answer
254
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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\...
- 3,209
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 ...
- 28k
0
votes
0
answers
151
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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
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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
1
vote
0
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114
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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
6
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5
answers
1k
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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 ...
- 1,805
0
votes
0
answers
122
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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)?
$$
My thoughts:
We know that $\mathrm{dim_H}(X)+\mathrm{dim_H}(...
- 63.9k
5
votes
1
answer
511
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Hausdorff dimension of boundaries of open sets diffeomorphic to $\mathbb{R}^n$
Let $B$ be a bounded open subset of $\mathbb{R}^n$ which is diffeomorphic to $\mathbb{R}^n$. (I am not sure how important the diffeomorphism is but this is the case I am interested in.) Let $C$ be its ...
CommunityBot
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4
votes
1
answer
969
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Usable Change-of-Variables Formula for Hausdorff Measure
Let $H^{s}$ be the $s$-dimensional Hausdorff measure, let $D$ be a nonsingular matrix. Consider the change of measure formula:
$$
\int\limits_{A} f(Dx) \; \mathrm{d}H^{s}(x) = \int\limits_{ D A} f(y)...
CommunityBot
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3
votes
0
answers
204
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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 $F\subset \mathbb{R}^m$
$$
\dim_HF+\dim_HE\leq \dim_H(E\times F)\leq \dim_HE+\...
CommunityBot
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8
votes
2
answers
388
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Isometrically-invariant measures and dilation of the Cantor set
Let $C$ be the Cantor middle-thirds set. Let $\mu$ be a finitely-additive isometrically-invariant measure on all subsets of $\mathbb R$. Then $\mu(3C)=2\mu(C)$, where $aB = \{ ax : x \in B \}$. ...
- 2,555
2
votes
1
answer
856
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Hausdorff dimension of a subset of Cantor set
What is the Hausdorff dimension of the subset
$$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 ...
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