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.
133 questions
4
votes
1
answer
221
views
Hausdorff dimension of graphs of singular functions
Let $f: \mathbb R^n \to \mathbb R^m$ be continuous, and differentiable almost everywhere with $Df = 0$ almost everywhere.
Question: What is the maximal Hausdorff dimension of the graph of $f$?
- 9,930
6
votes
2
answers
354
views
Can a Lipschitz function have derivative 0 on a dense set of small dimension?
Let $f: \mathbb R^n \to \mathbb R$ be Lipschitz continuous. Denote by $Z(f)$ the set on which $f$ is differentiable with derivative $0$.
Suppose $f$ is such that $Z(f)$ is topologically dense.
...
- 9,930
8
votes
1
answer
345
views
Hausdorff dimension of the stretch set of a Lipschitz map
Let $f: \mathbb R^n \to \mathbb R^m$ be a non constant Lipschitz map, for $m < n$. We denote by
$$\text{Lip}(f):= \sup_{x, y \in \mathbb R^n} \frac{|f(x) - f(y)|}{|x - y|}$$
the best Lipschitz ...
- 9,930
3
votes
1
answer
135
views
Hausdorff dimension of non-smooth, connected quadratic Julia sets
Q. Is is true that if $c$ is a parameter in the Mandelbrot set such that the corresponding Julia set $J_c$ of $z^2+c$ is not smooth, then the Hausdorff dimension of $J_c$ is greater than 1?
This seems ...
- 105
0
votes
0
answers
73
views
How to use tube union volume estimates to infer Kakeya conjecture
Wang Hong and Zahl's work "
Volume estimates for unions of convex sets, and the Kakeya set conjecture in three dimensions“ says
Why this is OK ? (We know Kakeya maximum function estimates can ...
- 839
3
votes
1
answer
108
views
Reference Request: accessible points of Wada domain boundaries in $\mathbb R^d$
Say that a compact $K\subset\mathbb R^d$ has the Wada property if there are disjoint, connected open sets $V_1,V_2,\ldots,V_n\subset\mathbb R^d$, $n\geq3$, such that $K=\partial V_1=\partial V_2=\...
- 221
2
votes
1
answer
208
views
Hausdorff dimension of the exceptional set of the gradient of an eikonal function
For $n \geq 2$, let $f: \mathbb R^n \to \mathbb R$ be everywhere differentiable, Lipschitz continuous, and an almost everywhere solution to the eikonal equation $|\nabla f| = 1$ a.e.
Question: What is ...
- 9,930
1
vote
0
answers
66
views
Hausdorff dimension of graph
Let $B_t$ be a one dimensional Brownian motion and $T\subset \mathbb{R}^+$ such that the Hausdorff dimension dim$T=1/2$. Is that possible to get the Hausdorff dimension of the following graph
$\{(B_t,...
- 31
5
votes
0
answers
101
views
Attainability of the conformal dimension of Sierpiński gasket
The conformal dimension of a metric space $(X,d)$ is defined as the infimum of the Hausdorff dimensions of all metric spaces quasisymmetric to $(X,d)$. A natural question is whether this infimum is ...
- 201
10
votes
1
answer
347
views
Can the set of non-differentiability of a Lipschitz function be of arbitrary Hausdorff dimension?
Let $n$ be a positive integer, and $s \leq n$ a positive real number.
Does there exist a Lipschitz function $f: \mathbb R^n \to \mathbb R$ such that the set on which $f$ is not differentiable has ...
- 9,930
1
vote
0
answers
136
views
When do solutions to $a^d+b^d=1$ correspond to self-similar sets?
Suppose $S$ is a set of Hausdorff dimension $d$, which admits a dissection into two pieces $S_a,S_b$. If the ratios of similarity between those pieces and the original set are $a$ and $b$ respectively,...
0
votes
0
answers
124
views
Partitioning fibers
Consider the following setup: A map $S:\mathbb{R}^d\to \mathbb{R}^{2m}$ which is Lipschitz and $d>m$. I have a probability density function $\rho$ on $\mathbb{R}^d$ for which I would like to ...
- 109
0
votes
0
answers
83
views
Finding a smooth curve whose intersection with Borel set has large dimension
Suppose $A\subset\mathbb{R}^2$ is a Borel measurable subset, and suppose $\dim_H A>a$ for some $a\in(0,1)$. Is it true that for any $\delta$, we can find a curve $\gamma$ (or even just require it ...
- 43
4
votes
1
answer
388
views
Cantor subset of a Borel set
Let $A\subset\mathbb{R}^n$ be a Borel measurable subset, then a classical result in descriptive set theory says that $A$ is either countable, or contains a Cantor subset $C$ (i.e. a subset ...
- 43
2
votes
1
answer
120
views
Maximal Hausdorff dimension of $p$-hollow points
Note: Here $\mu$ denotes the Lebesgue measure in $\mathbb R^n$.
Let $E$ be a measurable subset of $\mathbb R^n$. We say a point $x \in \mathbb R^n$ is $p$-hollow for $p > 1$, if $\mu(E \cap B_r (x))...
- 9,930