Questions tagged [dimension-theory]
Hausdorff dimension, box dimension, packing dimension and similar concepts.
0
votes
0answers
36 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
vote
0answers
75 views
Infinite Noetherian ring of dimension $1$ in which distinct non-zero ideals have distinct and finite index
Let $R$ be an infinite commutative ring with unity such that every non-zero ideal has finite index. Then $R$ is Noetherian, every non-zero prime ideal is maximal , and I can also show that $R$ is an ...
5
votes
1answer
228 views
A question on dominant morphism of affine schemes
Let $A \subseteq B$ be a ring extension where $A,B$ are both finitely generated $\mathbb C$-domain of the same Krull dimension. Also assume $A$ is regular (i.e. $A_{ \mathfrak p}$ is regular local ...
27
votes
1answer
769 views
If $\text{dim}(X \times X) = 2\text{dim}(X)$, does $\text{dim}(X^n) = n\text{dim}(X)$?
I have been learning some (topological) dimension theory and have gotten through most of the basic material, at this point, and am about to start looking at papers. In particular, I want to get ...
1
vote
1answer
83 views
Understanding equivalent condition for covering dimension
Let dim $X$ denote the Lebesgue covering dimension for a topological space $X$. Now a result in common books concerning dimension theory states the following:
If $X$ is a normal topological space, ...
4
votes
1answer
269 views
Submersion implies many rational points in image?
Let $A \colon V \to W$ be a surjective linear map
(defined over $\mathbb{Z}$),
inducing a projection
$\alpha \colon \mathbb{P}(V) \to \mathbb{P}(W)$.
Let $X \subseteq \mathbb{P}(V)$ and $Y \...
2
votes
0answers
38 views
Dimensions and volumes of continuous images of closed Riemannian manifolds
Suppose $M$ is an $m$-dimensional closed Riemannian manifold and $f \colon M \to \mathbb{R}^{n}$ is continuous. I'm interested in the case when $M=\mathbb{T}^{m}$. Let $\mathcal{M}_{f} := f(M)$. Let $...
2
votes
1answer
117 views
Topologically Ordered Families of Disjoint Cantor Sets in $I$?
Suppose that we have an uncountable collection $C_\alpha$ of disjoint Cantor Sets contained in the closed unit interval $I$. Suppose we have ordered the indices $\alpha \in [0,1]$ as well. Then is ...
3
votes
2answers
107 views
What is the dimension of a subspace of the product of $n$ linearly ordered compacta
This question is motivated by this problem of Dominic van der Zypen.
Problem. Let $X=\prod_{i=1}^nX_i$ be the Tychonoff product of linearly ordered compact Hausdorff spaces $X_1,\dots,X_n$. Is it ...
4
votes
0answers
92 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 ...
5
votes
1answer
277 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 ...
16
votes
1answer
349 views
Lowest Dimension for Counterexample in Topological Manifold Factorization
Bing gave a classical example of spaces $X, Y, Z$ such that $X \times Y = Z$, where $X$ and $Z$ are manifolds but $Y$ isn't. The space $Z$ in his example has dimension four. Is it known if this is ...
4
votes
1answer
206 views
Automorphisms of Erdös spaces
It is well-known that there is an automorphism of the Cantor set $h:C\to C$ such that $\overline{\{h^n(c):n\in \mathbb Z\}}=C$ for every $c\in X$.
In other words, there is a self-homeomorphism of $C$...
5
votes
1answer
153 views
Direct limit of Cantor sets
Let $C$ be the Cantor set, and $\omega$ the discrete space of integers $\{0,1,2,...\}$.
My conjecture:
(1) For each $n<\omega$ let $f_n:C\to C$ be a continuous function (possibly not onto). Let ...
4
votes
2answers
214 views
Ordinal vs. cardinal dimension
$\newcommand{\Ord}{\operatorname{Ord}}$When does it make sense to define the dimension of a space to be an infinite ordinal, instead of restricting to infinite cardinals?
We would essentially have to ...
1
vote
0answers
161 views
What intuitive concepts in pure math can be used to understand Big data? [closed]
I am not a mathematician but I need mathematicians' general knowledge and that is why I chose this community to ask my question from.
As a student/researcher in Data Mining, with background in Pure ...
4
votes
2answers
184 views
Doubling dimension vs other metric dimensions
For separable metric spaces, three fundamental notions of dimension
are equivalent:
$$ \text{dim }X = \text{Ind }X = \text{ind }X ,$$
Where does the doubling dimension
fit into the picture?
6
votes
1answer
101 views
Equi-Hölder embeddings of compact metric spaces of finite packing dimension into $\ell_2$
Problem. Does a compact metric space of finite packing dimension admit an equi-Hölder embedding into a Hilbert space?
A map $f:X\to Y$ between metric spaces $(X,d_X)$, $(Y,d_Y)$ is called equi-Hölder ...
5
votes
2answers
353 views
Can we define an isomorphism invariant to measure “dimension” of an undirected simple graph?
Can we define a characteristic to measure the "dimension" of a graph?
Let's start by some simple example.
Intuitively, a circuit graph with $n$ nodes and $n$ edges should have dimension $1$. ...
2
votes
0answers
77 views
Why is a hyperbolic basic set of dimension 2 either an attractor or a repeller?
I'm currently trying to understand the Birman-Williams Template Theorem, proved in the paper "Knotted periodic orbits II: Fibered knots, Low Dimensional Topology". Unfortunately, there doesn't seem to ...
9
votes
3answers
446 views
When is “metric dimension” well defined?
A subset $B$ of a metric space $(M,d)$ is called a metric generating set if and only if $$[\forall b \in B, d(x,b)=d(y,b)] \implies x = y \,. $$ A metric generating set $B$ is called a metric basis ...
0
votes
2answers
79 views
Growth of the size of coverings for sets with prescribed upper Minkowski dimension
For subsets of $\mathbb{R}^n$, I want a notion of dimension $\operatorname{dim}$ verifying:
If $\operatorname{dim}(A) = d$, then there's a constant $C$, depending only on $A$, $d$ and $n$, such ...
5
votes
0answers
115 views
On fractional dimensions with $\dim(A\times B)=\dim A+\dim B$
The Hausdorff dimension satisfies
$$ \dim A + \dim B \leq \dim (A\times B) $$
while the (upper) packing dimension has
$$ \dim_p (A\times B) \leq \dim_p A + \dim_p B $$
These inequalities hold for any ...
7
votes
2answers
169 views
Isoperimetric dimension for any (metric) measure space?
$\newcommand{\v}{\operatorname{vol}}$The isoperimetric dimension is the maximum $d$ s.t.
$$\v(D)\leq C\cdot \v(\partial D)^{d/d-1}$$
for all open with smooth boundary $D\subset M$, differentiable ...
0
votes
1answer
159 views
Visualizing the 4th dimension [closed]
In a freshers lecture of 3-D geometry, our teacher said that 3-D objects can be viewed as shadows of 4-D objects. How does this helps us visualize 4-D objects?
I searched that we can atleast see ...
1
vote
0answers
56 views
Measure for which lower Ledrappier (box, Minkowski) dimension and Hausdorff dimension disagree?
I assume that an example of the following is well known to experts, but I've had no luck finding it in the literature:
Given a measure $\mu$ on $\mathbb R^n$, let $\mathcal N(\mu, \epsilon, \delta)$ ...
2
votes
0answers
49 views
Number of Nearest-Neighbors in high dimensions [closed]
Consider $n$ adversarially chosen points in $\mathbb{R}^{d}$ where $n \gg d$. Let $\mathbf{a}$ be one of the $n$ points. Is there an upper bound on the number of points among the remaining $n-1$ ...
5
votes
1answer
269 views
Is there a gap between the Hausdorff and the lower Minkowski dimensions?
Does there exist a subset $A\subseteq\mathbb{R}^n$, for some $n$, and numbers $h<m$, such that the Hausdorff dimension $\dim A=h$, while for every cover $A_i$, $A\subseteq\bigcup_{i=1}^\infty A_i$ ...
2
votes
1answer
82 views
Fiber dimension formula for compact Hausdorff spaces?
In Algebraic Geometry one has a very useful formula for the dimension of fibers. Specifically I am thinking about a statement of the following form:
Let $C$ be a curve over $\mathbb{C}$, and let $S$...
2
votes
1answer
173 views
Extremally disconnectedness and 0-dimensional space
Let $X$ be a non-empty topological space. Then we have the following concepts for the topological space $X $:
1) We say $X $ has property $*$, if for every closed subset $A$ of $X$ and every open ...
3
votes
0answers
84 views
Dimension of a graph
Is it true that the graph of a function $\varphi:\mathbb [0,1]\to\mathbb R$ which is discontinuous at each $x$, has lower box dimension strictly greater than one?
If not, what extra condition do we ...
1
vote
0answers
60 views
Is left dimension preserved by left translation?
Let ${\bf K}\supset K\supset L$ be division rings with $[K:L]_{\rm left}=\infty$, and $a\in {\bf K}^\times$.
Question. Is it possible that $[aK:L]_{\rm left}<\infty$ in the sense that there would ...
2
votes
1answer
171 views
One-dimensional topological spaces
We know that all connected (not a singleton) subsets of $\mathbb{R}$ (with the usual topology) has no empty interior. This fact does not remains true for a general connected topological space with the ...
2
votes
1answer
82 views
Topological spaces with Lebesgue covering dimension 1
We know that all connected subsets of $\mathbb{R}$( with the usual topology) has no empty interior. I would like to know if this fact remains true for a general
connected topological space with the ...
2
votes
0answers
34 views
What is a precise definition of a twisted fibration of one fern over another?
I am curious if there is a notion of a "twisted fibration" of fractals. Since there are many classes of fractals, I'll ask specifically about L-systems.
How can we precisely define the twisted ...
9
votes
1answer
318 views
Box dimension of the set of Pisot numbers?
A Pisot number is an algebraic integer bigger than $1$ with all of its Galois conjugates having modulus less than $1$. The set of Pisot numbers is known to be countably infinite and is not dense in $(...
6
votes
5answers
652 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 ...
4
votes
1answer
208 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 ...
6
votes
0answers
138 views
What is the meaning of complex values/multiplicities in dimension spectrum?
If we have a manifold $M$ (say smooth, closed) it can be equipped with the Laplace operator $\Delta$. One can consider the function $\textrm{trace}(\Delta^{-s})$ where $s$ is complex parameter and $\...
6
votes
2answers
539 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 $...
6
votes
1answer
241 views
Random Cantor sets on the unit interval
Denote $A=\{0\}, B=\{0,1\}$. Then any subset of $\Omega:=\{A,B\}^{\mathbb N}$ is a continuum provided the number of $B$'s is infinite. We treat these as binary expansions of numbers in $[0,1]$.
For ...
5
votes
0answers
136 views
Do $G_\delta$-measurable maps preserve dimension?
This question (in a bit different form) I leaned from Olena Karlova.
Question. Let $f:X\to Y$ be a bijective continuous map between metrizable separable spaces such that for every open set $U\subset ...
8
votes
1answer
162 views
Can a smooth diffeomorphism of a Riemannian manifold have only positive Lyapunov exponents on a large set?
Let $M$ be a compact Riemannian manifold, $f: M \to M$ a diffeomorphism, and $\mu$ an ergodic measure for $M$. Suppose that the support of $\mu$ is not a finite set. Is it possible that all the ...
11
votes
1answer
356 views
Dimension in CW-approximation
The following question was something that came to my mind during my (unsuccessful) attempt at answering this MO-question.
Let $X$ be a topological space, and let $\tilde{X}\to X$ be a CW-...
11
votes
2answers
511 views
Two definitions of Lebesgue covering dimension
Maybe this question has already been considered here, but after a quick search I didn't find what I was looking for.
As I see, in the literature there are two different definitions of the ...
5
votes
1answer
272 views
Factorization of a certain map through a CW-complex
Suppose that $X$ is a paracompact Hausdorff space (e.g. a metric space) with $\dim X=n$ (the Lebesgue covering dimension). I want to find a proof (or a reference) that any (continuous) map $f: X \to K(...
8
votes
1answer
283 views
$U_q(\mathfrak{sl}_2)$ representations of “quantum dimension” zero
I'm reading up on quantum groups and their applications and I've come across a question I just can't find an answer to. I know about the basic representation theory of $U_q(\mathfrak{sl}_2)$ and I ...
2
votes
0answers
232 views
What is the connection between the Riemann Xi-function and n-sphere? [closed]
Riemann's Xi-function is defined as
$$\xi(s) = \pi^{-s/2}\ \Gamma\left(\frac{s}{2}\right)\ \zeta(s)$$
At the same time we have the following formulas for n-sphere's area and volume:
$$\begin{array}{...
3
votes
1answer
199 views
Sort shapes in 4 dimensions [closed]
I'm sure you know the baby game about sorting shapes by putting them in holes:
I'm wondering if such a game could exist in 4 dimensions? I imagine the shapes would have 4 dimensions and the holes 3 ...
4
votes
1answer
71 views
Bound for the generalised Rényi dimension of a measure
If $\mu$ is a measure on $\mathbb{R}^d$, and for each $r>0$ we let $\mathcal{M}_r$ denote the set of all ``cubes'' of the form $[j_1r,(j_1+1)r) \times \cdots \times [j_dr,(j_d+1)r)$ for $j_1,\ldots,...
