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

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systems of parameters vs. minimal “exhausting” systems in a Noetherian local ring

Hello, Probably this is a very easy question. Fix a Noetherian local ring $A$, and an $A$-module of finite type $M$. Lets call a system $ x_1 , \ldots , x_m \in \mathfrak{m}$ $M$-exhausting, if $M / ...
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Hausdorff Dimension of non-locally maximal hyperbolic sets

We're referencing Yakov Pesin's "Dimension Theory in Dynamical Systems" in an effort to compute the Hausdorff dimension of a particular invariant set $\Lambda$ of a hyperbolic toral automorphism. ...
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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 ...
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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 $\...
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Does the Hausdorff dimension depend on the L^p-norm?

A simple question from someone new to the field: In a metric space, the Hausdorff dimension of a subset is defined by covering the subset with $\epsilon$-balls and looking at how the number of ...
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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 this ...
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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 $\le n$?
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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 ...
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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 Hausdorff)...
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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, ...
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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, i....
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Hausdorff dimension of higher powers of the Mandebrot set ?

My third question about Shishikura's result : Shishikura (1991) proved that the Hausdorff Dimension of the boundary of the Mandelbrot set equals 2, in this paper1. The Mandelbrot set is defined by ...
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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 ...
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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 ...
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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 ...
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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 $L^\infty(X,\mu)$,...
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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 ...
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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 ...
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Dimension of tensor product of modules

$A\rightarrow B$ a ring homomorphism of Noetherian rings, where $A$ is local. $M$, $N$ finitely generated and nonzero $A$- and $B$- modules, respectively. Then I seem to get $\mbox{dim}_ {B}(M\...
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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 ...
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Converse of Principal Ideal Theorem

$(A, \mathfrak{m})$ a Noetherian local ring, $a\in\mathfrak{m}$ a zero divisor. Then is it true that $\mbox{dim}\ A/(a) = \mbox{dim}\ A$ ?
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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 ...
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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 ...
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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 ...
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how slow can the dimension of a product set grow?

Let us define the following "dimension" of a Borel subet $B \subset \mathbb{R}^k$: $\dim(B) = \min\{n \in \mathbb{N}: \exists K \subset \mathbb{R}^n, ~{\rm s.t.} ~ B \sim K\}$, where $\sim$ denotes "...
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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 $\...
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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 ...