# Questions tagged [dimension-theory-analysis]

For questions about topological or metric notions of dimension, including the Lebesgue, small and large inductive, Hausdorff, and packing dimensions, etc. and for questions regarding fractal geometry and analysis. Use [linear-algebra] instead for questions about the dimension of vector spaces. Also note [dimension-theory-algebra].

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### Embedded subshift in $[0,1]$ has dimension $0$?

I was wondering whether most embeddings of one-dimensional subshifts have zero Hausdorff dimension? Given a finite alphabet $\mathcal{A}= \{ 0,...,d-1 \}$ and $\Omega\subseteq \mathcal{A}^\mathbb{N}$, ...
1 vote
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### Is there a nice class of topological spaces satisfying the "finite sum theorem" for the small inductive dimension?

The small inductive dimension $\operatorname{ind}X\in\{-1,0,\ldots\,\infty\}$ of a topological space $X$ is defined inductively, as follows. If $X=\varnothing$, then $\operatorname{ind}X=-1$. If for ...
1 vote
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### Reverse Holder functions

For a function defined on $[0,1]\subset \mathbb{R}$, we call it reverse-Holder with exponent $\alpha >0$ if there exists some $c > 0$ such that $$|f(x) - f(y)| \geq c|x-y|^\alpha.$$ The ...
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### A 4-dimensional orange peeling problem: Can you disassemble a 4D hypersphere into a 3D object?

Where I am coming from: Similar to if you peel an orange, you cannot project the 2D surface of the Earth onto a 2D plane without curves. However, I’m pretty sure it’s possible to take the ...
73 views

### How to prove $V$ is a linear space and find $\dim V$?

Let $f(x)$ be a cubic real polynomial with a leading coefficient of $1$. Let $\alpha$ be all the complex roots of $f(x)$. $$V=\{y\,|\,y=g(\alpha),g(x) \text{ is a real polynomial}\}$$ Prove that $V$ ...
1 vote
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### What is the "content" of a box?

Let $(X,d)$ be a metric space. If set $A\subseteq X$, let $H^{\alpha}$ be the $\alpha$-dimensional Hausdorff measure on $A$, where $\alpha\in[0,2]$ and $\text{dim}_{\text{H}}(A)$ is the Hausdorff ...
56 views

### Intuition behind the Sum Theorem in dimension theory

In the Menger-Urysohn dimension, for any two subspaces $A, B$ of $X$, there is $$\operatorname{dim}(A\cup B)\leqslant \operatorname{dim}(A)+\operatorname{dim}(B)+1$$ However, if $A, B$ are closed ...
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### Link between Hausdorff measure of a set and Hausdorff measure of a sphere

I'm reading a proof given by Edward Szpilrajn found here and it says, among other things, that if $A$ is a non-empty set, and if $m\in\mathbb{N}$ are such that $$(*)\quad \mathcal{H}^{m+1}(A) = 0 ,$$ ...
1 vote
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### Hausdorff Dimension of an Intersection

Let $E$ and $F$ be two non-empty subsets of $\mathbb{R}^n$ for some $n \in \mathbb{N}$. I'm wondering if there's a simple formula or bound for the Hausdorff dimension of $E \cap F$ in terms of the ...
1 vote
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### Prove Ker$T=($Im$T^*)^\bot$ and (Ker$T^*$)$^\bot$ = Im$T$. Deduce $\dim$Im$T=\dim$Im$T^*$.

Let $V$ be a finite dimensional inner product space and $T:V\to V.$ Prove Ker$T=($Im($T^*))^\bot$ and $($Ker$T^*)^\bot$ = Im$T$. Deduce dimIm$T$ = dimIm$T^*$. I have written a proof but I'm very ...
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### Will the upper bound $b=2$ work in a simplicial proof of Brouwer's Fixed Point Theorem, Hurewicz' and Wallman's "Dimension Theory"?

Background: I am self studying Hurewicz' and Wallman's 'Dimension Theory'. My question concerns pp. 37-39 there within. The theorem A) below is equivalent to Brouwer's FPT, hence the title. I will ...
1 vote
32 views

### Does the Box-Counting Dimension always exist after applying a Lipschitz mapping?

When you have a bi-Lipschitz mapping and the Box-Counting Dimension of $A$ is defined, then the Box-Counting Dimension of $f(A)$ ist also defined (and the same as the B.-Dim. of $A$). We can also ...
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### How do I find the dimension of a space with a closed partition with empty interior?

I am not a mathematician (so please be kind :) but I need to understand this problem in relation to my work in a different field. Let $\mathcal X,\mathcal Y,\mathcal Z$ be topological spaces with sets ...