# Questions tagged [dimension-theory]

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

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### 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 ...
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### 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 $V_{j}(t)=V_{j}(t,\xi_{j})$...
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### $\varepsilon$-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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 (...
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### 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 ...
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### 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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### 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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### 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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### 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 ...
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### 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 $1$. ...
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### 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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### 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 ...
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 ...
### 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 ...