21
votes
Accepted
Why do almost all points in the unit interval have Kolmogorov complexity 1?
I'm not an expert on Kolmogorov complexity, but this does seem like a counting argument: for any fixed $\epsilon > 0$, there are only $\sum_{i = 1}^{(1-\epsilon)n} 2^i < 2^{(1-\epsilon)n+1}$ ...
- 2,621
20
votes
Accepted
A set whose Hausdorff dimension gradually changes?
I assume you want a set $A\subseteq [0,1]$ such that $\dim (A\cap [0,x])=x$ for all $x$. We can define $A_1$ by taking the union of a (Borel) subset of dimension $0$ of $[0,1/2]$ with a subset of ...
- 24.2k
14
votes
Accepted
Existence of subset with given Hausdorff dimension
First of all, $\dim_{H} (A) = \alpha$ iff $ H^k(A)=\infty$ for all $k<\beta$ and $H^k(A) = 0$ for all $k>\beta$. Then $H^\alpha(A) = \infty$ for all $\alpha \in (0,\beta)$.
If $A$ is closed ...
- 1,277
14
votes
Accepted
Is the complement of a zero-dimensional subset of the plane path-connected?
If the zero-dimensional set $X$ is not closed, then the answer is "no".
To construct a suitable example, take any open bounded neighborhood $U\subset\mathbb R^2$ of zero, whose boundary $\partial U$ ...
- 41.8k
12
votes
Accepted
Hausdorff dimension of the graph of an increasing function
Theorem 1. The Hausdorff dimension of the graph $\Gamma_f$ of $f$ equals $1$.
Proof.
Take a partition of $[0,1]$ by intervals of length $1/n$. Since the function is increasing you can cover the graph ...
- 28k
12
votes
Accepted
If $\text{dim}(X \times X) = 2\text{dim}(X)$, does $\text{dim}(X^n) = n\text{dim}(X)$?
As John Samples noted in his comment, Dranishnikov's Theory of cohomological dimension implies the positive answer to this problem for compact (even $\sigma$-compact) metrizable spaces. Namely, ...
- 41.8k
12
votes
Existence of subset with given Hausdorff dimension
The answer is yes under the additional assumption that the set is compact and I do not know what happens in the general case. The result is a consequence of the following one, see [1] and references ...
- 28k
12
votes
Is the complement of a zero-dimensional subset of the plane path-connected?
At least if $X$ is compact, the answer is yes. Indeed, by Corollary 2 of Theorem IV 3 in:
W. Hurewicz, H. Wallman, Dimension Theory. Princeton Mathematical Series, v. 4. Princeton University Press, ...
- 28k
12
votes
Accepted
Unknown work of Nöbeling on topological/Hausdorff dimension
So, the sought for paper is:
Nöbeling, G., Hausdorffsche und mengentheoretische Dimension, Ergebnisse math. Kolloquium Wien 3, 24-25 (1931).
And here is a ``translation" (to English and to modern ...
- 622
10
votes
Fractals of dimension zero
One example: the set of
Liouville numbers has Hausdorff dimension zero.
In number theory, a Liouville number is an irrational number $x$ with the property that, for every positive integer $n$, ...
- 41.1k
10
votes
Box dimension of the set of Pisot numbers?
The set $\mathcal P$ of all Pisot numbers is known to be closed (Salem). Its limit points $\mathcal P'$ are also known in $(1,2)$ (Talmoudi). The smallest element of $\mathcal P'$ is the golden ratio $...
- 2,125
9
votes
Accepted
Lowest Dimension for Counterexample in Topological Manifold Factorization
As asked Dusan Repovs (who is an expert in the theory of topological manifolds), and he sent me the following answer:
This is indeed best possible result, since whenever a product of two spaces is a ...
- 41.8k
8
votes
Fractals of dimension zero
If by a fractal you mean a self-similar set and if the corresponding iterated function system (IFS) satisfies the Open Set Condition (OSC), then the answer is no.
The Moran-Hutchinson formula gives ...
- 2,125
8
votes
Accepted
Doubling dimension vs other metric dimensions
In one direction, a rapidly branching tree will have very high doubling dimension, while having topological dimension $0$ (or $1$, if you include the edges). In another direction there is a bound, and ...
- 96.4k
8
votes
Dimension of a topological space equals the supremum of the dimension of open subsets in an open cover
We want to show that if $X$ has a chain $\varnothing\neq Z_0\subsetneq\dots\subsetneq Z_n$ of irreducible subsets, then some $U_i$ contains a chain of irreducible subsets of the same length.
Since $...
- 28.2k
8
votes
What is special to dimension 8?
Some special properties of dimension 8, in addition to the ones you identify:
Bernstein's problem holds up to dimension $n=8$. The only function of $\mathbb{R}^{n-1}$ whose graph in $\mathbb{R}^n$ ...
8
votes
What is special to dimension 8?
The following is too long to leave as a comment, so I post it as an answer.
At least for the cases of the Bernstein problem for minimal graphs, Simons' theorem on stable minimal cones, etc., there is ...
8
votes
Unknown work of Nöbeling on topological/Hausdorff dimension
It seems to be:
Nöbeling, G., Hausdorffsche und mengentheoretische Dimension, Ergebnisse math. Kolloquium Wien 3, 24-25 (1931). ZBL57.0749.02.
Google shows the first and sporadically the second page ...
- 31.5k
8
votes
Accepted
Sequences with 0's in $\mathbb R ^\omega$
For every $n\in\mathbb N$ consider the space
$$X_n=\{x\in\mathbb R^\omega:\lvert x^{-1}(0)\rvert\ge n\}.$$
Theorem. For any positive integer numbers $n<m$, the spaces $X_n$ and $X_m$ are not ...
- 41.8k
7
votes
Ordinal vs. cardinal dimension
A classic text is Dimension Theory by Hurevicz & Wallman (1941)
Among the types of dimension for topological spaces are the "upper inductive dimension" and the "lower inductive dimension". ...
- 41.1k
7
votes
Accepted
Transitive homeomorphisms of Erdős spaces
The answer to both questions is affirmative.
Theorem 1. The complete Erdos space $\mathfrak E_c$ has a self-homeomorphism whose every orbit is dense in $\mathfrak E_c$.
Proof. We use a known result ...
- 41.8k
7
votes
Fraction dimensional "Euclidean" space
One remark: there is no metric space $\mathbb{E}^{1/2}$ of Hausdorff dimension $\frac{1}{2}$ whose topological square $\mathbb{E}^{1/2} \times \mathbb{E}^{1/2}$ is homeomorphic to $\mathbb{E}^1$, i.e.,...
- 344
7
votes
Accepted
Does finite Hausdorff dimension imply finite packing dimension?
A construction used (repeatedy) in the paper
Edgar, G. A., Centered densities and fractal measures, New York J. Math. 13, 33-87 (2007). ZBL1112.28004.
For more information, see that paper.
We ...
- 41.1k
6
votes
Fractals of dimension zero
Closed uncountable sets of Hausdorff dimension zero turn up from time to time in practice (one of my papers with Nikita Sidorov features one) but I think that there does not exist a set which is ...
- 6,206
6
votes
Accepted
Is there a gap between the Hausdorff and the lower Minkowski dimensions?
I believe what you are asking is whether the Hausdorff dimension can be strictly less than the lower packing dimension (also called the lower modified box dimension). I'm pretty sure the strongest ...
- 3,312
6
votes
Two definitions of Lebesgue covering dimension
Yes, the long ray $R$ works. If $\mathcal{U}$ is a finite open cover then $\bigcap\{R\setminus U:U\in\mathcal{U}\}=\emptyset$ and at least one of these closed sets must be bounded as in $R$ any two ...
- 11.4k
6
votes
Accepted
Real rank 0 implies stable rank 1 on $C^\ast$-algebras?
There are "cheap" examples. If a C*-algebra $A$ has stable rank 1 then it is stably finite. On the other hand every simple, unital purely infinite C*-algebra has real rank 0. So every simple, ...
- 2,689
6
votes
Accepted
Box dimension of the graph of an increasing function
Pietro Majer's argument that you cited actually shows that the upper box dimension is $1,$ and hence the lower box, the upper and lower packing, and the Hausdorff dimensions are all equal to $1.$ Also,...
- 3,312
6
votes
Accepted
Can you remove a zero dimensional subspace from a cube and obtain a planar space?
I think the answer is negative in any dimension $n\geq 2$.
Theorem:
$\mathbb R^n$ can not be covered by a zero-dimensional set and a set homeomorphic to a subspace of $\mathbb R^{n-1}$.
Proof:
Suppose ...
- 770
6
votes
Accepted
Prove that $ n \leq d+1 $ under ordering constraints in $\mathbb{R}^d$
For $d=2$ the maximal $n$ is $3=d+1$. Indeed, between 4 points $x_1,x_2,x_3,x_4$ either one of them, $x_k$, belongs to the convex hull of three others, then $\langle \theta,x_k\rangle$ can not be the ...
- 109k
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