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

**7**

votes

**3**answers

334 views

### How can dimension depend on the point?

Let $M$ be a metric space.
For any subset $A\subset M$ let $\dim(A)$ denote its Hausdorff dimension.
For $x\in M$, define the dimension of $M$ at $x$ by $\dim(x)=\lim_{r\to0}\dim(B(x,r))$; this limit ...

**10**

votes

**1**answer

286 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 ...

**6**

votes

**1**answer

211 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

**1**answer

238 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 ...

**3**

votes

**0**answers

69 views

### Is the doubling dimension invariant under Möbius maps?

Is the doubling dimension of a metric space $(X,d)$ invariant under maps $f:(X,d) \to (X,d')$, which are the identity on $X$ and which map the metric $d$ to a metric $d'$ which has the same cross ...

**8**

votes

**1**answer

168 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 ...

**1**

vote

**0**answers

123 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:
...

**0**

votes

**0**answers

74 views

### Distortion of the Hausdorff dimension of sums of Cantor sets under local scaling

The following question deals with possible distortion of the Hausdorff dimension of sums of Cantor sets as one "zooms in" on the sum around any given point.
Let us assume that $C_1$ and $C_2$ are two ...

**4**

votes

**1**answer

60 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 ...

**13**

votes

**3**answers

2k views

### Dimension of subalgebras of a matrix algebra

If $n$ is given and $A$ is a subalgebra of $M_n(\mathbb C)$, the algebra of $n \times n$ matrices with entries in the field of complex numbers, then what are the possible values of dimension of $A$ as ...

**3**

votes

**1**answer

104 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 ...

**10**

votes

**3**answers

532 views

### Self-Similar Graphs

Many fractals can be generated using and infinite sequence of graphs. For example, Sierpinski's Gasket could be generated by the following sequence of graphs.
Many definitions of fractal dimensions ...

**5**

votes

**1**answer

132 views

### Multifractal Analysis and Dimension Spectrum of Unions

Consider the classical Multifractal Analysis, and the decomposition of the state space $X$ into level sets
$$X=\bigcup_{\alpha}\left\{x\mid d_\mu(x)=\alpha\right\}\cup\left\{x\mid d_\mu(x) ...

**22**

votes

**2**answers

1k views

### Analogues of Luzin's theorem

If $X$ is a compact metric space and $\mu$ is a Borel probability measure on $X$, then the space $C(X)$ of continuous real-valued functions on $X$ is a closed nowhere dense subset of ...

**3**

votes

**0**answers

210 views

### dimension of induced comodule

Let $\pi : G \to H$ be epimorphism of Hopf superalgebras, where $G$ be an quantum super group of function on $GL(m|n)$, $H$ be an quantum group of function on $GL(m) \otimes GL(n)$; $W$ an finite ...

**3**

votes

**2**answers

657 views

### How big can the Hausdorff dimension of a function graph get?

This question is inspired by How kinky can a Jordan curve get?
What is the least upper bound for the Hausdorff dimension of the graph of a real-valued, continuous function on an interval? Is the ...

**4**

votes

**1**answer

99 views

### Existence of small projective dimensioned modules

Suppose $A$ is a (if necessary unital) associative ring and $I$ is a left ideal in $A$. Let $\operatorname{pd}(M)$ denote the projective dimension of a left $A$-module $M$.
Then do either of the ...

**20**

votes

**3**answers

2k views

### Krull dimension <= transcendence degree?

Let $k$ be a field, and $A$ a $k$-domain, so that the fraction field of $A$ has transcendence degree $n$ over $k$.
If $A$ is finitely-generated over $k$, then $A$ has Krull dimension $n$ (Theorem A ...

**6**

votes

**1**answer

383 views

### Geometric measures different from Hausdorff

$\newcommand{\RR}{\mathbb{R}}\newcommand{\calF}{\mathcal{F}}\newcommand{\diam}{\mathrm{diam}}$
In geometric measure theory there are various notions of $m$-dimensional measure for sets $A\subset ...

**15**

votes

**2**answers

512 views

### A “dimension” for Tychonoff spaces

It's well-known that any Tychonoff space $X$ can be embedded in $[0,1]^k$ for some cardinal $k$. It's natural to ask what the smallest such $k$ is (let's call it $k(X)$). However, this probably ...

**1**

vote

**0**answers

224 views

### Global dimension of a subalgebra with all units

(All rings here are always assumed to be unital and associative).
Setup
Let $R$ be a ring, and $A$ and $B$ be $R$-algebras, with $A$ a commutative subalgebra of $B$ satisfying:
If $u$ is a unit ...

**12**

votes

**3**answers

473 views

### Dimensions of self-affine sets

Let $A$ be a $2\times 2$ matrix which we assume to be contracting, i.e., the exists $\alpha\in(0,1)$ such that
$$
\|A {\mathbf x}\|_2\le \alpha\|{\mathbf x}\|_2,\quad \forall {\mathbf x}\in\mathbb ...

**1**

vote

**1**answer

304 views

### Hochschild cohomology and formal smoothness

Hochschild cohomology can be used to characterise formal smoothness of unital associative algebras; in that such an algebra $A$ is formally smooth if and only if it is of Hochschild cohomological ...

**1**

vote

**3**answers

923 views

### Zero-dimensional space

Let $X$ be a topological space with the following property: for any open subset $A$ of $X$ there is a collection of clopen subsets $\{A_{\alpha} : \alpha\in S\}$ such that ...

**2**

votes

**0**answers

151 views

### Is there a better function (linear or even a projection)?

Let $A$ be a finite $n$-element set. Let $\mathbb R^A$ be an $n$-dimensional Euclidean space (with the ordinary Euclidean distance). Let $X$ be an arbitrary topological space. Consider a continuous ...

**13**

votes

**2**answers

713 views

### infinite dimensional CAT(0) groups

Usually a CAT(0) group is defined to be a group acting properly isometrically and cocompactly on a CAT(0) space, but I would like to consider only those groups that act properly, isometrically and ...

**8**

votes

**1**answer

641 views

### When does the homological dimension of a tensor product equal the sum of dimensions?

The notion of dimension I prefer most is right global dimension, but the question can also be asked for other notions (e.g. weak dimension, injective dimension, Krull dimension). Letting $d$ be ...

**10**

votes

**1**answer

222 views

### Why do convex polytope options constrict with dimension, rather than expand?

There are an infinite number of regular polygons in the plane,
five regular polyhedra,
six regular polytopes in $\mathbb{R}^4$,
and then three regular polytopes in every dimension $d > 4$.
There ...

**12**

votes

**1**answer

5k views

### Rank of a module

What's wrong with defining the rank of a finitely generated module over any (commutative) ring to be just the smallest number of generators? All books I know define rank only locally this way. But why ...

**4**

votes

**2**answers

547 views

### Do constructible sets have Krull dimension?

Let $(I,\leq)$ be a poset. Recall that the Krull dimension of $I$ is defined as follows:
-- $K.dim(I)=-1$ if and only if $I=\{0\}$;
-- if $\alpha$ is an ordinal and we already defined what it means ...

**4**

votes

**1**answer

256 views

### Lebesgue dimension of closures satisfying the Z-set condition

Given any subspace $A\subset X$ of a topological space with Lebesgue dimension $\le N$.
Let $\bar{A}$ denote the closure of $A$. Assume, that the pair $(\bar{A},A)$ satisfies the Z-set condition, ...

**16**

votes

**3**answers

3k views

### Nonseparable example in dimension theory?

Could you give me an example of a complete metric space with covering dimension $> n$ all of which closed separable subsets have covering dimension $\le n$?
The question closely related to ...

**5**

votes

**1**answer

149 views

### Is there a survey of recent work relating to the Hausdorff dimension of sets defined through some restriction of digits?

I am familiar with the work of Helmut Cajar, but his book is thirty years old and it's clear that there has been substantial progress since then. I have been spending a lot of time looking through ...

**3**

votes

**0**answers

113 views

### Possible Hausdorff dimension of intersection of Besicovitch-Eggleston like sets

Let $b \geq 2$ be an integer and suppose that $v=(p_0,\cdots,p_{b-1})$ be a probability vector. Let $S_{b,v}$ be the set of real numbers whose $b$-ary expansion has the digit $k$ with relative ...

**6**

votes

**1**answer

356 views

### Arithmetic products of Cantor sets.

Let $A,B\subseteq \mathbb{R}$ be two Cantor sets. What is known about the arithmetic product
$AB=\lbrace ab|a\in A, b\in B\rbrace$? In particular, what is known in the case that the sets are ...

**2**

votes

**2**answers

143 views

### A notion of a 'coarse', parametrized dimension of an object, where the parameter determines how finely we can distinguish (say) a very thin rod from a line

I apologize for the clumsy wording of the title-- what I'm looking for is a notion of an integer-valued dimension $d_{\epsilon}$, which we parametrize by a real positive number $\epsilon$, of, say, a ...

**18**

votes

**6**answers

1k views

### Pathological Examples of Dimension

I am trying to wrap my head around all the different notions of dimension (and their equivalences). To get a sense of this, it would be nice to know the subtle difficulties that arise. I thus think it ...

**0**

votes

**0**answers

106 views

### When does the rank of a module behave sub-multiplicatively under tensoring?

Let $\cal{E}$ be a finitely generated projective bimodule over a (noncommutative) algebra $A$. Moreover, let us assume that $\cal{E}$ is of finite rank $n$. The tensor product
$
\cal{E} \otimes_A ...

**4**

votes

**2**answers

180 views

### Hutchinson's formula for asymptotically homogeneous Cantor sets

As everyone knows, the standard middle-thirds Cantor set is constructed by dividing the interval into three equal parts, removing the middle one, then applying the same procedure to the remaining two ...

**11**

votes

**6**answers

2k views

### Different definitions of the dimension of an algebra

I know of three ways to define the dimension of a finitely-generated commutative algebra A over a field F:
The Gelfand-Kirillov (GK) dimension, based on the growth of the Hilbert function.
The Krull ...

**2**

votes

**1**answer

122 views

### Can the isoperimetric dimension of a d-generated group attain any value?

Background
The isoperimetric dimension of a finitely generated group $G$, which we denote by $\dim(G)$, is the largest number $d$ such that any Cayley graph $\Gamma$ of $G$ (constructed with respect ...

**6**

votes

**2**answers

402 views

### On some finiteness properties for schemes

Consider the following properties of scheme $X$:
A: $X$ is of finite type over $\mathbb{Z}$
B: $X$ is Noetherian
C: $X$ is of finite Krull dimension
What implications are there between these ...

**7**

votes

**4**answers

542 views

### Tessellating $\mathbb{R}^n$ by bricks.

Let us call the $\ell_1$-product of intervals $[0,k_1]\times...\times [0,k_n]$ a brick of size $k_1+...+k_n$. Consider a tessellation $T$ of $\mathbb{R^n}$ by (shifted) bricks so that every point ...

**11**

votes

**5**answers

940 views

### Is there an axiomatic approach of the notion of dimension ?

There are many notions of dimension : algebraic, topological, Hausdorff, Minkowski... (and others).
While the topological one generalize the algebraic one, the last three need not coincide for every ...

**0**

votes

**1**answer

329 views

### existence of fractal [closed]

I have a question about fractals;
Suppose $\alpha\in[0,1]$ is real number, is there any fractal $F_\alpha$, such that $Dimension(F_\alpha)=\alpha$?
If yes, do we have any method to construct such ...

**4**

votes

**3**answers

686 views

### dimension of a real affine variety

Let $V$ be a real affine variety in $\mathbb R^n$, i.e. the zero set of a real polynomial $p(x_1,\dots,x_n)$. Consider the following three definitions of the dimension of $V$, $dim(V)$.
Definition ...

**4**

votes

**3**answers

401 views

### minimum space dimension to place n-points knowing pairwise distances

Hi everyone,
Let P be a set of n points.
Assuming I know the pairwise distances for each pair of points.
What would be the minimum dimension of the space in which I could place those n points with ...

**5**

votes

**3**answers

423 views

### Quantitative measurement of infinite dimensionality

I recently encountered the metric mean dimension, which is a numerical metric invariant of (discrete time, compact space) dynamical systems that refines topological entropy for infinite-entropy ...

**2**

votes

**2**answers

635 views

### Simple definition of the Hausdorff measure using squared paper

I am giving a "non-technical" seminar in which I would like to give an elementary introduction to the Hausdorff dimension and measure.
For simplicity, I was hoping to give a more intuitive ...

**0**

votes

**0**answers

188 views

### Decompose a set into sets of Hausdorff-dimension n-1

Assume we can decompose a set $A$ in $\mathbb{R^n}$ of Hausdorff-dimension n into sets $(A_t)$ $t\in [0,1]$ of Hausdorff-dimension n-1 whose n-1-dimensional volume is known (for example is zero).
...