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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.

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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 ...
Noah Schweber's user avatar
1 vote
1 answer
109 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( (-\...
Karl Fabian's user avatar
3 votes
0 answers
117 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 ...
Matheus Manzatto's user avatar
5 votes
0 answers
90 views

A generalized Hausdorff dimension in form of a Lower semi continuous function

Let $(X,d)$ be a compact metric space. Assume that $f:X\to \mathbb{R}$ is a positive continuous function. We say that the $f$-dimension of $(X,d)$ is equal to $0$ if for every $\epsilon>0$ ...
Ali Taghavi's user avatar
2 votes
1 answer
147 views

Hausdorff dimension of the non-differentiability set of a locally Lipschitz function

Let $f:\mathbb R^n \to \mathbb R$ and $E := \{x \in X : f \text{ not Fréchet differentiable at }x\}$. Then $E$ is Borel measurable. It is well-known that Theorem If $f$ is convex, then the Hausdorff ...
Akira's user avatar
  • 713
2 votes
1 answer
89 views

Hausdorff dimension and non-empty intersections with lines

Let $A\subseteq [0,1]^d$, $d\geq 2$, a set with Hausdorff dimension $\operatorname{dim}_{\mathcal{H}}A=s$. What is the minimum $s$ (if any) which guarantee that $A$ has non-empty intersections with a ...
RaffaeleScandone's user avatar
6 votes
1 answer
280 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 ...
Peter's user avatar
  • 213
3 votes
2 answers
194 views

Hausdorff dimension of the non-differentiability set a convex function

Let $X \subset \mathbb R^d$ be open, $f : X \to \mathbb R$ and $$ E := \{x \in X : f \text{ is not Fréchet differentiable at }x\}. $$ Then we have the following result which is Theorem: If $X= \...
Akira's user avatar
  • 713
1 vote
0 answers
82 views

Lower bound estimate for the sum $\sum \text{diam}(U)^d$ over all countable covers of a cube

This question is inspired from the definition of Hausdorff measure. Let $C$ be a closed unit hypercube in $\mathbb R^d$ (side length equal to one, including boundary. The cube itself is at top ...
No One's user avatar
  • 1,531
6 votes
1 answer
371 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 ...
No One's user avatar
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4 votes
1 answer
169 views

On the set on which $|Df|$ is maximal for Lipschitz $f$

Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz continuous function with strict Lipschitz constant $L > 0$. That is, $|f(x) - f(y)| < L|x - y|$ for all $x \neq y$ in $\mathbb R^d$. Question: ...
Nate River's user avatar
  • 2,574
5 votes
1 answer
262 views

Hausdorff measure

Let $(X,d)$ be a metric space and let $H^\alpha$ denote the $\alpha$-dimensional Hausdorff measure on $X$, where $\alpha$ is the Hausdorff dimension of $X$. Is there any simple condition on $X$ that ...
Wreck it Ralph's user avatar
1 vote
0 answers
105 views

Hausdorff dimension of a compact Lie group [closed]

Let $G$ be a compact Lie group (for simplicity assume $G=SO(3)$). Equip $G$ with a left-invariant Riemannian metric and let $m$ be left-invariant Haar measure on $G$. Now that $G$ is a metric space ...
Adam's user avatar
  • 323
2 votes
1 answer
328 views

Fraction dimensional "Euclidean" space

The “dimension” of Euclidean space $\mathbb{E}^n$ can be explained as an algebraic property, simply as a dimension of a vector space over the field $\mathbb{R}$. It also can be understood as a ...
Kacper Kurowski's user avatar
2 votes
0 answers
147 views

Relationship between Hausdorff dimension and covering number

Let $(X,d)$ be a compact metric space and recall that the $\epsilon$-external covering number $\mathcal{N}^{\epsilon}(X)$ of $X$ is defined by: $$ \mathcal{N}^{\epsilon}(X) := \inf\left\{ N\in \mathbb{...
ABIM's user avatar
  • 4,881
5 votes
1 answer
567 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 ...
Claude's user avatar
  • 605
4 votes
1 answer
223 views

Hausdorff dimension and critical exponent of words

What is the Hausdorff dimension of the subset $S_c \subset [0,1]$ of points such that the critical exponent of their binary expansion is $c$? It's clear that $\dim_H S_{\infty}=1$, but what can be ...
Alessandro Della Corte's user avatar
3 votes
2 answers
275 views

Maximal Hausdorff dimension of the set on which derivatives do not agree

Let $f, g: [0, 1] \to \mathbb R$ be functions that are differentiable a.e. with $f’ = g’$ almost everywhere. What is the maximal Hausdorff dimension $d$ (and corresponding Hausdorff $d$-measure) of ...
Nate River's user avatar
  • 2,574
4 votes
2 answers
306 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 ...
matthew's user avatar
  • 51
3 votes
1 answer
458 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: ...
rfloc's user avatar
  • 431
5 votes
0 answers
532 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 ...
Joe's user avatar
  • 759
4 votes
1 answer
231 views

The Hausdorff codimension of singular matrices vs. the Hausdorff codimension of points with divergent trajectories

Let $G:=SL(m+n,\mathbb R)$ and $\Gamma :=SL(m+n,\mathbb Z)$ and $X:=G/\Gamma$. (1) Let $M$ denote the set of all $m \times n$ matrices with real entries. A matrix $A \in M$ is called $\textit{singular}...
No One's user avatar
  • 1,531
2 votes
0 answers
201 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 ...
katago's user avatar
  • 533
2 votes
1 answer
120 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 ...
Geoffrey Irving's user avatar
2 votes
0 answers
85 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 ...
user6419's user avatar
  • 411
3 votes
0 answers
76 views

For $\mathcal{L}^1$-a.e. $t\in R$, Hausdorff dimension of level sets of a locally Lipschitz function $f:R^n\to R$ is $n-1$?

Let $f:R^n\to R$ be a locally Lipchitz function. Denote $H^n$ the n-dimensional Hausdorff measure. We know that for any $H^n$-measurable subset $A\subset R^n$, for $\mathcal{L}^1$-a.e. $t\in R$, $A\...
mathmetricgeometry's user avatar
2 votes
0 answers
43 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 ...
user109433's user avatar
3 votes
2 answers
966 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, ...
Anixx's user avatar
  • 8,838
2 votes
0 answers
88 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 ...
truebaran's user avatar
  • 8,736
5 votes
0 answers
112 views

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}^...
Dirk's user avatar
  • 11.9k
6 votes
1 answer
197 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 $\beta$?
Nicola Arcozzi's user avatar
2 votes
0 answers
140 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 ...
user48633's user avatar
1 vote
1 answer
184 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 ...
Ali Taghavi's user avatar
11 votes
0 answers
298 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 ${\rm II_1}$ ...
Sebastien Palcoux's user avatar
2 votes
0 answers
63 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 ...
QCD_IS_GOOD's user avatar
3 votes
1 answer
213 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 ...
Riku's user avatar
  • 729
7 votes
1 answer
243 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 ...
Severin Schraven's user avatar
22 votes
3 answers
957 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 ...
Severin Schraven's user avatar
2 votes
1 answer
186 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\...
Thomas Yang's user avatar
11 votes
1 answer
832 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 $$ \...
Piotr Hajlasz's user avatar
13 votes
1 answer
519 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 $\...
Gro-Tsen's user avatar
  • 26.4k
0 votes
0 answers
103 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?
Dopax's user avatar
  • 1
2 votes
0 answers
71 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 ...
ljjpfx's user avatar
  • 175
6 votes
3 answers
464 views

About the Hausdorff dimension of removable singularities of PDE

There are some interesting phenomenons about removable singularities (or extension problems). In the theory of functions of several complex variables, we know the classical Hartogs theorem: Let $f$ ...
user128943's user avatar
2 votes
0 answers
74 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-...
DavidHume's user avatar
  • 681
7 votes
1 answer
189 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 $I$ ...
Jacob Denson's user avatar
4 votes
0 answers
111 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 ...
Jörg Neunhäuserer's user avatar
7 votes
1 answer
997 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 ...
Nikita Sidorov's user avatar
1 vote
0 answers
108 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 ...
Guo's user avatar
  • 11
3 votes
1 answer
154 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=\{...
Dominik Kwietniak's user avatar