Range of the Hausdorff dimension

In summary, the conversation discusses how the range of Hausdorff dimension is all nonnegative real numbers and how to prove this. It is suggested that in \mathbb{R}^n, there is no set with dimension greater than n. To construct a set with dimension between 0 and n, one can use the example of Cantor-like sets in \mathbb{R} and adapt it to \mathbb{R}^n by starting with a closed box and deleting everything except the corners. Further hints are needed to determine the dimension of this set.
  • #1
OB1
25
0
My analysis textbook mentioned in passing that the range of the Hausdorff dimension is all nonnegative real numbers, i.e. for any nonnegative real number a, there's some compact subset of R^n whose Hausdorff dimension is exactly a. The problem is that I don't see how to prove this (and my oh-so-concise book doesn't bother proving it). Does anyone know how to go about proving this?
 
Physics news on Phys.org
  • #2
in [itex]\mathbb{R}^n[/itex], you will not find any set that has a dimension [itex]>n[/itex] ! To construct a set with Hausdorf dimension [itex]0 < d < n[/itex], it suffices to construct a set [itex]E[/itex] that has a [itex]d[/itex]-Hausdorf measure [itex]0 < \mathcal{H}^{(d)}(E) < \infty[/itex]. If you know the example of Cantor-like sets in [itex]\mathbb{R}[/itex], you can adapt the idea in [itex]\mathbb{R}^n[/itex].
 
Last edited:
  • #3
So, suppose I start with the closed box in [tex] \mathbb{R}^{n}[/tex] and delete everything except the corners of the box with side length [tex] l [/tex], and then repeat this so the ith iteration leaves boxes of side length [tex] l^{i}[/tex]. So I have a Cantor-like set in [tex] \mathbb{R}^{n}[/tex]. I'm still a bit puzzled on finding the dimension of this object, any further hints?
 

1. What is the Hausdorff dimension?

The Hausdorff dimension is a mathematical concept used to measure the "size" or "dimension" of a set in a metric space. It was first introduced by the German mathematician Felix Hausdorff in the early 20th century.

2. How is the Hausdorff dimension different from the topological dimension?

The topological dimension of a set is a discrete whole number, while the Hausdorff dimension can be any real number between 0 and infinity. The topological dimension is also based on the concept of open sets, while the Hausdorff dimension is based on the concept of covering a set with smaller sets.

3. What is the significance of the Hausdorff dimension in mathematics?

The Hausdorff dimension has many applications in various fields of mathematics, including fractal geometry, dynamical systems, and probability theory. It is also used in the study of self-similar structures and chaotic systems.

4. How is the Hausdorff dimension calculated?

The Hausdorff dimension of a set is calculated by taking the limit of the ratio of the logarithm of the number of smaller sets needed to cover the set to the logarithm of the size of those smaller sets. This limit is often referred to as the "Hausdorff measure".

5. Can the Hausdorff dimension be greater than the topological dimension?

Yes, the Hausdorff dimension can be greater than the topological dimension. This is because the Hausdorff dimension takes into account the "thickness" or "irregularity" of a set, while the topological dimension only considers its discrete dimensionality.

Similar threads

  • Topology and Analysis
Replies
2
Views
1K
Replies
1
Views
3K
Replies
1
Views
343
  • Calculus and Beyond Homework Help
Replies
1
Views
453
Replies
5
Views
1K
Replies
2
Views
1K
  • Beyond the Standard Models
Replies
1
Views
1K
  • Set Theory, Logic, Probability, Statistics
Replies
1
Views
732
Replies
3
Views
805
Back
Top