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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What is $3/2$-dimensional Hausdorff measure of the graph of Brownian motion?
It is well-known, and well-documented, that the Hausdorff dimension of the graph of regular $1$-dimensional Brownian motion is $3/2$ (almost surely). See for example Theorem 4.29 in "Brownian ...
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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 ...
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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 $K$ ...
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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}...
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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 ...
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What is the Hausdorff dimension of the set on which this exponential sum is bounded?
This is a direct follow up to For which rationals is this exponential sum bounded?
Given $x \in [0, 1]$, we denote by $e(x)$ the complex number $e^{2 \pi i x}$.
What is the Hausdorff dimension of the ...
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How large can the set of turbulent points be?
This question resisted attempts on MSE.
Let $E \subset \mathbb R^n$ be a Lebesgue measurable set. We say that $x \in \mathbb R^n$ is a turbulent point of E if both the following conditions hold:
$$\...
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Is the $L^\infty$ norm of the derivative the same under the Hausdorff and Lebesgue measure?
Note: Here $\mathcal H^k$ denotes the $k$-dimensional Hausdorff measure, and $\|f\|_{L^\infty (\mathcal H^k)}$ denotes the $L^\infty$ norm of a function $f$ with respect to $\mathcal H^k$.
Let $\Omega$...
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For $\mathcal{L}^1$-a.e. $t\in\mathbf R$, is $n-1$ the Hausdorff dimension of level sets of a locally Lipschitz function $f:\mathbf R^n\to\mathbf R$?
Let $f:\mathbf R^n\to\mathbf R$ be a locally Lipchitz function. Denote $\mathrm H^n$ the $n$-dimensional Hausdorff measure. We know that for any $\mathrm H^n$-measurable subset $A\subset\mathbf R^n$, ...
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Is there an explicit, everywhere surjective $f:\mathbb{R}\to\mathbb{R}$ whose graph has zero Hausdorff measure in its dimension?
Suppose $f:\mathbb{R}\to\mathbb{R}$ is Borel. Let $\text{dim}_{\text{H}}(\cdot)$ be the Hausdorff dimension, and $\mathcal{H}^{\text{dim}_{\text{H}}(\cdot)}(\cdot)$ be the Hausdorff measure in its ...
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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\...
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Hausdorff dimension of the zero set of the gradient of an eikonal function
Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz function with $|\nabla f| = 1$ almost everywhere with respect to Lebesgue measure.
What is the supremal Hausdorff dimension of the set on which $f$ is ...
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Hausdorff dimension of the zero set of $\nabla f$
Let $f: \mathbb R^n \to \mathbb R$ be a Lipschitz function with $\nabla f$ nonzero almost everywhere with respect to Lebesgue measure.
What is the supremal Hausdorff dimension of the set on which $f$ ...
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Hausdorff dimension of Apollonian circle packing, 1.305686729, 1.305688 or yet something else?
I am interested in the Hausdorff dimension of the Apollonian circle packing.
There seem to be two numerical calculations of the value:
1.305686729(10)
from P.B ...
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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 supremal Hausdorff dimension of the set on which $f$ and $g$ are both ...
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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 $\...
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Estimating the Hausdorff dimension of the discontinuity set of a function
Suppose $\sum_{\xi \in \mathbb{Z}^d}{a_{\xi}}e^{i\langle x, \xi\rangle }$ converges spherically pointwise to $0$ for all $x \in \mathbb{T}^d$, i.e. $\lim_{R \to \infty} \sum_{|\xi| < R}{a_{\xi}}e^{...
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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 ...
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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 ...
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Kaplan-Yorke dimension when all Lyapunov exponents are positive or the system is 1-dimensional
The Lyapunov/Kaplan-Yorke dimension $D_{KY}$ can be calculated using the Lyapunov Exponents of an n-dimensional dynamical system, as shown in the relevant Scholarpedia entry.
If $\lambda_1>\...
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Hausdorff dimension of the curve of a continuous nowhere differentiable function
It is of course well-known that there are plenty of functions from $\mathbb R$ into itself which are continuous and nowhere differentiable. Although the Baire Category Theorem is enough to prove the ...
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The Hausdorff dimension of the set of reals of inner models
Suppose that both $M$ and $N$ are models of $ZFC$ with $M\subseteq N$ so that $M$ is definable in $N$.
Question Can $(\mathbb{R})^M$ have Hausdorff dimension strictly between $0$ and $1$ in $N$? How ...
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Steinhaus theorem and Hausdorff dimension
Assume for simplicity that sets $A_i\subset\mathbb{R}$ are compact. If $A_1$ and $A_2$ have positive length, then $A_1+A_2$ contains an interval. That is a variant of the classical Steinhaus theorem ...
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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}^...
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Finding a unique and finite expected value for almost all measurable functions?
Let $(X,d)$ be a metric space. If set $A\subseteq X$, let $H^{\alpha}$ be the $\alpha$-dimensional Hausdorff measure on $A$, where $\alpha\in[0,+\infty)$ and $\text{dim}_{\text{H}}(A)$ is the ...
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Are the extensions of the expected value, below, finite for all functions in only a shy subset of all measurable functions?
This is a follow up to this post, where I wish to verify whether one of the statements (in the post) is true but first let's recap the definitions:
Let $(X,d)$ be a metric space. If set $A\subseteq X$,...
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Finding examples of functions which are infinite or undefined with current extensions of the expected value?
Preliminaries
Consider the expectations desribed in this paper, which is an extension of the Lebesgue density theorem; this paper which is an extension of the Hausdorff measure, using Hyperbolic ...
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Defining a measure of uniformity for measurable subsets of $[0,1]^2$ w.r.t dimension $\alpha\in[0,2]$
Let $(X,d)$ be a metric space. If set $A\subseteq X$, let $H^{\alpha}$ be the $\alpha$-dimensional Hausdorff measure on $A$, where $\alpha\in[0,2]$ and $\text{dim}_{\text{H}}(A)$ is the Hausdorff ...
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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 ...
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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 ...
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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( (-\...
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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 ...
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How big can the Hausdorff dimension of a function graph get?
This question is inspired by How kinky can a Jordan curve get?
What is the least upper bound for the Hausdorff dimension of the graph of a real-valued, continuous function on an interval? Is the ...
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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 $\epsilon>0$ ...
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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 ...
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Best Hölder exponents of surjective maps from the unit square to the unit cube
The Peano's square-filling curve $p:I\to I^2$ turn's out to be Hölder continuous with exponent $1/2$ on the unit interval $I$ (a quick way to see it, is to note that $p$ is a fixed point of a ...
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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= \...
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Hausdorff dimension of convex set in ${\bf R}^n$
I want to know the smoothness of convex set in ${\bf R}^n$.
Recall the following definition.
Definition : $X$ is a bounded closed convex set in ${\bf R}^n$ if for $x$, $y\in X$, the any $d$-...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $f$ ...
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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: ...
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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 ...
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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 ...
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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 ...
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Construction of null sets with prescribed Hausdorff dimension and generalizations
Given $h:\mathbb{R}_0^+ \to \mathbb{R}_0^+$ increasing and right continuous, the outer measure $\mathcal{H}^h$ in $\mathbb{R}^d$ that assigns to every $E\subset\mathbb{R}^d$ the measure $$\mathcal{H}^...
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Relationship between Hausdorff dimension and covering number
Let $(X,d)$ be a compact metric space and recall that the $\epsilon$-external covering number $\mathcal{N}^{\epsilon}(X)$ of $X$ is defined by:
$$
\mathcal{N}^{\epsilon}(X) := \inf\left\{
N\in \mathbb{...
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