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

**33**

votes

**3**answers

3k views

### What the heck is the Continuum Hypothesis doing in Weibel's Homological Algebra?

On page 98 of Weibel's An Introduction to Homological Algebra he mentions that the ring $R = \prod_{i=1}^\infty \mathbb{C}$ has global dimension $\geq 2$ with equality iff the continuum hypothesis ...

**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 ...

**19**

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 ...

**17**

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 ...

**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 ...

**15**

votes

**2**answers

504 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 ...

**14**

votes

**2**answers

1k views

### Area of the boundary of the Mandelbrot set ?

My second question about Shishikura's result :
Shishikura (1991) proved that the Hausdorff Dimension of the boundary of the Mandelbrot set equals 2, in this paper 1. In a sense, could we consider it ...

**13**

votes

**2**answers

680 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 ...

**12**

votes

**3**answers

2k views

### Dimension of subalgebras of a matrix algebra

If n is given and A is a subalgebra of M_n(C), the algebra of n-by-n matrices with entries in the field of complex numbers, then what are the possible values of dimension of A as a vector space over ...

**12**

votes

**3**answers

396 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 ...

**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 ...

**11**

votes

**3**answers

683 views

### Why do modules with small support have high Exts?

Let $M$ be a module over a ring $R$. In nice situations (though I don't know what exactly nice means...) the following two numbers are equal:
1.) The codimension of the support of $M$
2.) The ...

**11**

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 ...

**11**

votes

**5**answers

917 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 ...

**10**

votes

**3**answers

446 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 ...

**10**

votes

**1**answer

219 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 ...

**10**

votes

**1**answer

411 views

### Can dividing out a group action can increase the Lebesgue dimension ?

Given any space $X$ of Lebesgue dimension at most $n$. Suppose a group $G$ acts on $X$ continuously. Can the dimension of the quotient $G\backslash X$ exceed the dimension of $X$?
I know examples, ...

**7**

votes

**2**answers

497 views

### Non-Hölder continuous devil's staircases

Let $f:[0,1]\to[0,1]$ be a devil's staircase in the usual sense. (That is, $f$ is continuous, non-decreasing, $f'=0$ on a set of full Lebesgue measure.) We also require the complement to the set where ...

**7**

votes

**4**answers

537 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 ...

**7**

votes

**1**answer

605 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 ...

**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 ...

**6**

votes

**3**answers

891 views

### Can we say anything about the Krull dimension of a localization?

I'm looking for a theorem of the form
If $R$ is a nice ring and $v$ is a reasonable element in $R$ then Kr.Dim$(R[\frac{1}{v}])$ must be either Kr.Dim$(R)$ or Kr.Dim$(R)-1$.
My attempts to do ...

**6**

votes

**5**answers

2k views

### More upper/lower semi-continuous functions in (algebraic) geometry?

The notion of upper/lower semi-continuity is sometimes encountered in algebraic geometry.
Here by upper semi-continuity one means a function on a topological space $f:X\rightarrow S$ with value in ...

**6**

votes

**1**answer

337 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 ...

**6**

votes

**1**answer

324 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 ...

**5**

votes

**3**answers

417 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 ...

**5**

votes

**1**answer

147 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 ...

**5**

votes

**1**answer

119 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) ...

**4**

votes

**2**answers

540 views

### Converse of Principal Ideal Theorem

$(A, \mathfrak{m})$ a Noetherian local ring, $a\in\mathfrak{m}$ a zero divisor. Then is it true that $\mbox{dim}\ A/(a) = \mbox{dim}\ A$ ?

**4**

votes

**3**answers

395 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 ...

**4**

votes

**2**answers

305 views

### How indepenedent of a chosen metric is the box-counting dimension? Is there a non-integral dimension which is defined for topological spaces?

Question 1. Given a topological space $X$ and two metrics $a$ and $b$ on it, compatible with the topology, what conditions should I impose on them so that box-counting (or other, for example ...

**4**

votes

**1**answer

697 views

### Example in dimension theory

Could you give me an example of a complete metric space wiht covering dimension $> n$ all of which compact subsets have covering dimension $\le n$?

**4**

votes

**4**answers

627 views

### Determining a lower bound on the Hausdorff dimension of a set

Does anyone know of a good method for finding a lower bound of the Hausdorff dimension of a set $G$?
The only method I could find is to find an $\alpha$-Hölder function $f \colon G \to H$ then ...

**4**

votes

**3**answers

660 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

**1**answer

97 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 ...

**4**

votes

**2**answers

525 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

683 views

### Topological dimension, is it local?

Let $n\in\mathbb N$ and $X$ be a complete metric space.
Assume that there is $\epsilon>0$ such that
$$\dim B_\epsilon(x)\le n$$
for any $x\in X$.
Is it true that $\dim X\le n$?
Here ...

**4**

votes

**2**answers

176 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 ...

**4**

votes

**1**answer

254 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, ...

**4**

votes

**1**answer

2k views

### Dimension of module

Does dimension of a module (say, dimension of its support) have anything to do with the supremum length of chains of prime submodules like rings?
Let's restrict to finitely generated modules over ...

**3**

votes

**2**answers

629 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 ...

**3**

votes

**1**answer

669 views

### Hausdorff dimension of graphs .

Is there an easy way to calculate the Hausdorff dimension of the graph of a real "elementary" function, like $f(x)=\sin(1/x)$ ?

**3**

votes

**2**answers

246 views

### Hausdorff dimension of inverse images.

Let $f: \mathbb{R}^d \to \mathbb{R}$ be a continuous function. Let $t \in (\inf(f), \sup(f))$ and define $C = f^{-1} (t)$. Is it true that the Hausdorff dimension of C is $\geq d -1$? If no how does ...

**3**

votes

**2**answers

294 views

### Gaps in Dimension Polynomials

There are several notions of rank/dimension defined on differential fields. However, we do not have a reasonable way to estimate these typically ordinal valued invariants. Especially, we do now know a ...

**3**

votes

**0**answers

107 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 ...

**2**

votes

**2**answers

612 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 ...

**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 ...

**2**

votes

**1**answer

258 views

### Hausdorff dimension of subsets of the Mandelbot set.

Shishikura (1991) proved that the Hausdorff Dimension of the boundary of the Mandelbrot set equals 2, in this paper, but I can't figure out one thing : can we say all open subsets of this boundary has ...

**2**

votes

**2**answers

141 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 ...

**2**

votes

**0**answers

149 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 ...