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

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4
votes
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97 views

Zero-dimensional functions in the plane

Is the following true? Conjecture. Let $\varphi:C\to [0,\infty)$ be an upper semi-continuous function, where $C\subseteq \mathbb R$ is a Cantor set. Let $X$ be a zero-dimensional subset of the graph ...
0
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0answers
22 views

What is the analog of the symmetrized Jacobi matrix for delay equations?

For a linear system of ODEs in $\mathbb{R}^{n}$ (with the usual inner product), say $\dot{V}(t) = A(t) V(t)$, we know that if $\xi_{1},\ldots,\xi_{k} \in \mathbb{R}^{n}$ and $V_{j}(t)=V_{j}(t,\xi_{j})$...
4
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1answer
111 views

$\varepsilon$-net of a $d$-dimensional unit ball formed by power set of $V = \{+1, 0 -1\}^d$

I have a set of $d$-dimensional vectors $V = \{+1, 0, -1\}^d $. Then $P(V)$ constitutes the power set of $V$. I now construct a set of unit vectors $V_{\mathrm{sum}}$ from the power set $P(V)$ such ...
6
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2answers
340 views

Unknown work of Nöbeling on topological/Hausdorff dimension

Let $\mathcal{H}^n$ denote the Hausdorff measure, $\dim_H X$ the Hausdorff dimension, and $\dim X$ the topological dimension of $X$. A well known result of Szpilrajn (He changed his name to ...
24
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1answer
1k views

What is special to dimension 8?

Dimension $8$ seems special, as the partial list below might indicate. Is there any overarching reason that dim-$8$ is "more special" than, say, dim-$9$? Surely it isn't it, in the end, simply because ...
1
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1answer
74 views

Perfect images of complete Erdős space

Let $\mathbb P$ denote the space of irrational numbers. In an answer to this question, Taras Banakh showed that the perfect images of $\mathbb P$ are precisely the Polish spaces with no compact ...
2
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1answer
169 views

Quotients of the irrationals

Everyone knows that there is a closed equivalence relation $\sim$ on the Cantor set $C$ such that each non-trivial equivalence class has exactly $2$ points and $[0,1]\simeq C/\sim$. Thus a closed ...
2
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0answers
68 views

Fourier dimension of radial set

In his 1967 article "Sur un theoreme de R. Salem", Gatesoupe proved that if a set $A\subset [0,1]$ has Fourier dimension $\alpha$ then the set $\tilde A:=\{x\in \mathbb{R}^n: |x| \in A\}$ has Fourier ...
4
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0answers
410 views

The “core” of complete Erdős space

This question is about the Erdős spaces: $\mathfrak E=\{x\in \ell^2:x_n\in \mathbb Q\text{ for all $n<\omega$}\}$; and $\mathfrak E_c=\{x\in \ell^2:x_n\in \mathbb P\text{ for all $n<\omega$}\}...
10
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2answers
749 views

Is the complement of a zero-dimensional subset of the plane path-connected?

Let $X$ be a zero-dimensional subset of the plane $\mathbb R ^2$. Is $\mathbb R ^2\setminus X$ necessarily path-connected? I feel the answer must be yes but I need a reference. If it helps, assume $...
5
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1answer
85 views

A classification of $G_{\delta\sigma}$ zero-dimensional spaces?

Among separable metrizable spaces: Cantor set is the unique compact zero-dimensional space without isolated points. $\mathbb Q$ is the unique countable space without isolated points $\mathbb R \...
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87 views

Real Rank of $M_n(A)$

The real rank for C*-algebras was defined by Brown-Pedersen in [1] as a noncommutative analog of covering dimension. Given a unital C*-algebra $A$, its real rank $\mathrm{rr}(A)$ is the smallest ...
3
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2answers
853 views

A set whose Hausdorff dimension gradually changes?

Can there be a set whose Hausdorff dimension gradually changes? For instance, a set of real numbers contained in an interval, whose Hausdorff dimension is 0 at the beginning and 1 closer to the end, ...
10
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1answer
659 views

Why are homeomorphism groups important?

For a compact metric space $X$ let $\mathcal H(X)$ denote the set of homeomorphisms in the compact-open topology (also generated by sup metric). It is known that $\mathcal H(X)$ is a Polish ...
3
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139 views

Separating a countable closed set from a point

Let 𝑋 be a separable metrizable space. An intersection of clopen subsets of $X$ is called a C-set. Question. If each singleton of $X$ is a C-set, $A\subseteq X$ is closed and countable, $x\in X\...
6
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1answer
374 views

Is there an almost strongly zero-dimensional space which is not strongly zero-dimensional

A Tychonoff space $X$ is called strongly zero-dimensional if each functionally closed subset $F$ of $X$ is a $C$-set, which means that $F$ is the intersection of a sequences of clopen sets in $X$. A ...
4
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1answer
185 views

One-dimensional compacta as projective limits

Let $X$ be a (not necessarily metrizable) Hausdorff compact space of covering dimension = 1. Is it possible to express $X$ as a filtering projective limit of finite graphs? Here finite graphs ...
3
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0answers
90 views

Dimension of Alexandrov space which is homeomorphic to a manifold

Let $M^n$ be a smooth manifold of dimension $n$. Let $M$ given a metric with curvature bounded below in the sense of Alexandrov which induces the original topology of $M$. It is true that the ...
0
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1answer
212 views

Dimension of a topological space equals the supremum of the dimension of open subsets in an open cover

For a topological space $X$ which is covered by a family of open subsets $\{U_i\}$, show that $\dim(X)=\sup (\dim(U_i))$. I understand that $\dim(X)\geq \sup(\dim(U_i))$, so it only suffices to show ...
2
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0answers
37 views

Examples of essentially sub-linear functions

A dimension function is an increasing, continuous function $% f:\mathbb R_{+}\rightarrow \mathbb R_{+}$ such that $f(r)\to 0$ as $r\to 0$. Say that a dimension function $f$ is essentially sub-linear ...
0
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1answer
68 views

Hausdorff outer measure is finite if $\sum_{j = 1}^m |f(x_i)-f(x_{i-1})|^s \le c$ [closed]

Let $f:[0,1] \to \mathbb{R}, G = graph(f)$. If $\sum_{j = 1}^m |f(x_i)-f(x_{i-1})|^s \le c$ for all partitions $0 = x_0< \ldots < x_m = 1 $ then $H^s(G) < \infty$ What technique can I ...
3
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0answers
245 views

Covering lemmas in Hochman's ''On self-similar sets with overlaps and inverse theorems for entropy''

I am confused about the covering lemmas in the captioned work and really hope to get some ideas here. Firstly it is lemma 3.7. (Image of Lemma 3.7) (for convenience here is the lemma of this lemma (...
6
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2answers
173 views

Box dimension of the graph of an increasing function

This Hausdorff dimension of the graph of an increasing function shows that: Let $f$ be a continuous, strictly increasing function from $[0,1]$ to itself with $f(0)=0, f(1)=1$. Then $dim_H \; G = ...
22
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3answers
699 views

Existence of subset with given Hausdorff dimension

Let $A\subseteq \mathbb{R}$ be Lebesgue-measurable and let $0<\alpha<1$ be its Hausdorff dimension. For a given $0<\beta <\alpha$ can we find a subset $B\subset A$ with Hausdorff ...
3
votes
2answers
345 views

Topological dimension of the image of continuous surjective functions

Consider two topological spaces $(X,\tau)$ and $(Y,\omega)$ and a continuous surjective function $f\colon X\to Y$. Let $\mathrm{dim}(X)$ and $\mathrm{dim}(Y)$ denote the Lebesgue covering dimension ...
4
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1answer
171 views

Graded Grothendieck Group and Hilbert Polynomial

I was wondering if any of the arguments from elementary dimension theory of local noetherian rings could be simplified with knowledge of the Grothendieck group. Let $A$ be a noetherian graded $K$-...
2
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1answer
130 views

Real rank 0 implies stable rank 1 on $C^\ast$-algebras?

A $C^\ast$ algebra has defined stable rank (https://www.univie.ac.at/nuhag-php/bibtex/open_files/2079_Rieffel-StableRank.pdf) and real rank (https://core.ac.uk/download/pdf/82123484.pdf), which are ...
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64 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
102 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
306 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 ...
32
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1answer
848 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
101 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
290 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
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0answers
43 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
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1answer
121 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
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2answers
116 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
102 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 ...
6
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1answer
563 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
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1answer
395 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 ...
6
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1answer
336 views

Automorphisms of Erdős spaces

A surjective homeomorphism $h:X\to X$ is minimal if $\overline{\{h^n(x):n\in \mathbb N\}}=X$ for every $x\in X$. In other words, the orbit of each point is dense. Does either of the Erdös spaces $\...
5
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1answer
169 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
447 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
199 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
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2answers
255 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
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1answer
109 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
417 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 $1$. ...
3
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0answers
84 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
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3answers
520 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
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2answers
82 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
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0answers
135 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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