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Questions tagged [dimension-theory]

Hausdorff dimension, box dimension, packing dimension and similar concepts.

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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?
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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
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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
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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
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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
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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 \...
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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 $...
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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 ...
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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
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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
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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 ...
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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
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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
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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 ...
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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 ...
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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?
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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
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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
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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
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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
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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
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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 ...
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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 ...
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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
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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
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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
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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
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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 $(...
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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
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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 $\...
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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
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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 ...
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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 ...
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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 ...
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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-...
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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
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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
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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}{...
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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
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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,...