Questions tagged [hausdorff-dimension]
Questions about dimensions of possibly highly irregular or "rough" sets, Hausdorff–Besicovitch dimension and related concepts such as box-counting or Minkowski–Bouligand dimension.
115 questions
8
votes
1
answer
867
views
Fubini's theorem for Hausdorff measures
$B\subset \mathbb{R}^2$ is a Borel set. Define the slices $B_x:= \{y \in \mathbb{R}: (x,y) \in B \}$.
If $\lambda$ denotes the Lebesgue measure on $\mathbb{R}$, presentations of Fubini's theorem often ...
- 675
4
votes
2
answers
432
views
Hausdorff dimension of Julia set
Can anyone show me the proof "Hausdorff dimension of Julia set is strictly positive"?
For purpose to prove this we might have to prove the green function of basin of attraction to infinity ...
- 6,548
5
votes
0
answers
586
views
On the Hausdorff dimension of a Cantor set
In what follows I refer to this paper by Orevkov.
I am writing a paper on this, so if somebody is interested we could consider to write a joint paper.
Consider a sequence $R=\{R_n\}_n$ of strictly ...
- 779
4
votes
1
answer
905
views
Hausdorff dimension and surface measure
Could someone please indicate me some reference that contains the proof of the following theorem?
Below $\mathcal{H}^n$ denotes the $n$-dimensional Hausdorff outer measure in $\mathbb{R}^n$.
Theorem: ...
- 2,247
2
votes
0
answers
222
views
Is Kakeya conjecture open with some additional regularity condition on Kakeya map?
in the paper, ON KAKEYA MAPS WITH REGULARITY ASSUMPTIONS, the author of the paper consider the Kakeya conjecture with some regularity on Kakeya map,
Kekeya conjecture[Hausdorff dimension version]
If ...
- 543
11
votes
1
answer
963
views
Coarea inequality, Eilenberg inequality
The general statement of the coarea inequality known also as the Eilenberg inequality is:
Theorem. If $f:X\to Y$ is a Lipschitz map between metric spaces and $A\subset X$, $0\leq m\leq n$, then $$
\...
- 28k
2
votes
1
answer
158
views
Natural way to thicken Brownian motion to 2D?
If we have a smooth plane curve (Hausdorff dimension 1), we can thicken it by a small amount to get a 2D set (all points within distance $\epsilon$ to the curve).
What if we start with the graph of a ...
- 14.2k
2
votes
0
answers
89
views
Bounds on convergence of two orbits in the limit set of a Schottky group
Suppose we have two points in the limit set of a Schottky group, $x,y\in \Lambda(\Gamma)$. Consider the orbits of those points under a primitive subset $\Gamma'$ of $\Gamma,$ that is, one that does ...
- 441
2
votes
0
answers
48
views
(1, 2) stability and Hausdorff dimension
Recall that a set $E \subset \mathbb{R}^n$ is called (1,2) stable if it satisfies the following:
$$
W_0^{1,2}(E) = W_0^{1,2}(E^0),
$$
where $E^0$ is the interior of $E$ and $W_0^{1,2}(E)$ is defined ...
- 93
23
votes
3
answers
1k
views
Existence of subset with given Hausdorff dimension
Let $A\subseteq \mathbb{R}$ be Lebesgue-measurable and let $0<\alpha<1$ be its Hausdorff dimension.
For a given $0<\beta <\alpha$ can we find a subset $B\subset A$ with Hausdorff ...
- 622
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, ...
- 1,329
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
6
votes
1
answer
205
views
Subspaces of metric spaces having prescribed dimension
Let $(X,d)$ be a metric space having Hausdorff dimension $\alpha>0$ and let $0<\beta<\alpha$. Is there a metric subspace of $X$ having Hausdorff dimension $\beta$?
- 14.2k
2
votes
0
answers
267
views
Examples of essentially sub-linear functions
A dimension function is an increasing, continuous function $%
f:\mathbb R_{+}\rightarrow \mathbb R_{+}$ such that $f(r)\to 0$ as $r\to 0$.
Say that a dimension function $f$ is essentially sub-linear ...
- 13k
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
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 ...
- 28k
2
votes
0
answers
66
views
Universal structure of fractal spaces
In the same way that we can say manifolds are made of pieces that look like $\mathbb{R}^n$, is there any way to say that spaces with the same hausdorff dimension are made up of pieces that look the ...
- 187
3
votes
1
answer
230
views
Hausdorff dimension and $W^{1,1}$ functions
What can be said about the Hausdorff dimension of the image of a set by a $W^{1,1}$ map?
In other words,
what is the relationship between
$\mathrm{dim}_H f(A)$ and $\mathrm{dim}_H A$, where $f \in ...
- 1,277
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
272
views
Hausdorff dimension of the boundary of fibres of Lipschitz maps
Let $f: \mathbb{R}^m\rightarrow \mathbb{R}^{m-k}$ be a Lipschitz map.
Can we get a uniform estimate on the Hausdorff dimension of the boundaries of fibres of $f$? I.e. do we have an upper bound for
...
- 3,486
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\...
- 3,209
7
votes
1
answer
1k
views
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
13
votes
1
answer
577
views
Are Hausdorff measures on the real line Haar measures for some locally compact topology?
For $0\leq d\leq 1$, let $\lambda_d$ be the $d$-dimensional Hausdorff measure on $\mathbb{R}$. Note that it is translation-invariant. Does there exist a locally compact topology $\mathscr{T}_d$ on $\...
- 42k
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
83
views
The Hausdorff dimensions of variations of Jarnik sets
For $\alpha, \beta>3,$ define
$$\{(x,y)\in[0,1]\times [0,1]: \|qx\|\le q^{1-\alpha}, \|qy\|\le q^{1-\beta} \quad \text{for infinitely many $ q\in \mathbb{N}$}\}.$$
This set can be regarded as a two ...
- 4,713
1
vote
1
answer
148
views
Dimension of quotient of compact totally disconnected group action
Assume that $X$ is a compact metric space and $G$ is compact
totally disconnected group. And $X$ has isometric free $G$-action
i.e. $gx=x\Rightarrow g=e$.
Then the following holds $${\rm dim}\ ...
CommunityBot
- 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
4
votes
0
answers
119
views
A quantity that distinguishes finer than Hausdorff dimension
Consider sets $A\subseteq \mathbb{R}$ with Lebesgue measure zero and Hausdorff dimension one. For instance the set of real numbers with bounded entries in their continued fraction expansion have ...
- 366
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
4
votes
1
answer
177
views
Measures maximizing entropy in a set of measures with fixed average for some observable
Let $\Omega$ be the set of all infinite binary sequences $(x_i)_{i\ge 0}$ endowed with the product topology coming from discrete topology on $\{0,1\}$.
Consider $0<\alpha<1$ and let $$K_\alpha=\{...
- 8,903
2
votes
0
answers
73
views
Closed set containing infinite arithmetic progressions of ANY gap
Let $A\subseteq [0,\infty)$ be a set containing infinite arithmetic progressions of ANY gap, that is, for any $d>0$, there is $t>0$ such that $t+kd\in A$ for all $k\in \mathbb N$.
Molter and ...
- 751
6
votes
5
answers
1k
views
Fractals of dimension zero
Are there any famous examples of fractals, or other closed sets, of cardinality continuum but Hausdorff dimension 0?
I can think of something ad hoc like a Cantor middle $\frac13$ set where the ...
- 1,805
1
vote
0
answers
251
views
Compact sets of Hausdorff dimension zero
I have a question about Hausdorff dimension. Suppose S is a compact subset of $\mathbb{R}^n$ whose Hausdorff dimension is zero. Does it follow that S can be covered by a finite DISJOINT union of ...
- 11
0
votes
0
answers
122
views
Hausdorff dimension of $X\times X$
I am thinking of the following question:
Let $X\subseteq \mathbb R$. Is it true that
$$
\mathrm{dim_H}(X\times X)=2\mathrm{dim_H}(X)?
$$
My thoughts:
We know that $\mathrm{dim_H}(X)+\mathrm{dim_H}(...
- 63.9k
3
votes
1
answer
114
views
volume entropy and Hausdoff dimension
In relation to this question: Relation between volume entropy and Hausdorff dim of limit set?
Given a metric space $Z$ and a hyperbolic approximation $X := hyp_{r_0}(Z)$ (as defined for example here)....
- 447
9
votes
1
answer
638
views
Is there a characterization of the Hausdorff measures?
It is known that there is a unique measure on the Borel $\sigma$-algebra of $\mathbb{R}^n$ such that the measure of the rectangle $\prod_i [a_i,b_i[$ is $\prod_i (b_i-a_i)$. This is the Lebesgue ...
- 702
7
votes
1
answer
494
views
Jarník-Besicovitch and outer measure
The set $A_\tau$ of irrational numbers $x$ which are $\tau$-approximable, i.e., that satisfy the estimate
$$\left|x - \frac{p}{q}\right| \leq \frac{1}{q^\tau}$$
for infinitely many rationals $p/q$, ...
- 295
5
votes
1
answer
511
views
Hausdorff dimension of boundaries of open sets diffeomorphic to $\mathbb{R}^n$
Let $B$ be a bounded open subset of $\mathbb{R}^n$ which is diffeomorphic to $\mathbb{R}^n$. (I am not sure how important the diffeomorphism is but this is the case I am interested in.) Let $C$ be its ...
CommunityBot
- 1
7
votes
2
answers
2k
views
Haar measure on the Grassmannian space
The grassmannian space $G(n,m)$ may be identified with the quotient space $O(n)/(O(m)\times O(n-m)$. As such, it is endowed with a natural invariant probability measure which I call "Haar measure on $...
- 26.3k
4
votes
1
answer
196
views
Is there a concept of uniform Hausdorff dimension?
Let $M$ be a metric space and let $U \subset M$ be open. Then the Hausdorff dimension of $U$ is defined in the usual way. If there is a single dimension number $d$ that is the Hausdorff dimension of ...
- 66
4
votes
2
answers
2k
views
Hausdorff dimension vs. cardinality
What is the relationship between the Hausdorff dimension and cardinality of a set?
Specifically, assuming the Continuum Hypothesis, if a set has Hausdorff dimension greater than zero does, that imply ...
- 4,713
4
votes
0
answers
255
views
Controlling the size of the balls in Hausdorff dimension/measure
Let $X$ be a compact metric space, and let
$$
\nu_s(X):=\sup\limits_{\varepsilon>0} \inf\limits_{\mathcal{E}} \sum\limits_{E \in \mathcal E} \mathrm{diam}(E)^s
$$
be the $s$-dimensional Hausdorff ...
- 41
3
votes
0
answers
199
views
Hausdorff Dimension of Exceptional Set for Carleson's Theorem
In Mattila's book Fourier Analysis and Hausdorff Dimension, Mattila presents a result of Barcelo, Bennett, Carbery, and Rogers about convergence of solutions of the Schrodinger equation to the initial ...
- 31
4
votes
1
answer
969
views
Usable Change-of-Variables Formula for Hausdorff Measure
Let $H^{s}$ be the $s$-dimensional Hausdorff measure, let $D$ be a nonsingular matrix. Consider the change of measure formula:
$$
\int\limits_{A} f(Dx) \; \mathrm{d}H^{s}(x) = \int\limits_{ D A} f(y)...
CommunityBot
- 1
25
votes
2
answers
3k
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 $\mathit{...
- 428
4
votes
2
answers
279
views
Hausdorff dimension of sequence space
Let $\Omega =\{0,1\}^{\mathbb{N}}$ denote the set of infinite sequences with elements $0$ or $1$. Let $d$ be the metric on $\Omega$ given by $d((x_n),(y_n))=1/2^m$, where $m=\min\{i\in\mathbb{N}\,:\,...
- 83
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+\...
CommunityBot
- 1
7
votes
3
answers
679
views
How can dimension depend on the point?
Let $M$ be a metric space.
For any subset $A\subset M$ let $\dim(A)$ denote its Hausdorff dimension.
For $x\in M$, define the dimension of $M$ at $x$ by $\dim(x)=\lim_{r\to0}\dim(B(x,r))$; this limit ...
CommunityBot
- 1
9
votes
1
answer
1k
views
When is Hausdorff measure a Frostman measure?
Let $(X,d)$ be a metric space and let $\mathcal{H}^s$ be the $s$-dimensional Hausdorff measure on $X$.
For a measure $\mu$ on $X$, we say that $\mu$ is a Frostman measure (sometimes referred as ...
CommunityBot
- 1
3
votes
1
answer
181
views
Packing measure and Kleinian groups
There has been "some" debate on the notion of fractal (as an illustration, see for example the discussion in this link). One of the possible notions includes relating Hausdorff dimension and packing ...
CommunityBot
- 1