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Results tagged with hausdorff-dimension
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user 11009
Questions about dimensions of possibly highly irregular or "rough" sets, Hausdorff–Besicovitch dimension and related concepts such as box-counting or Minkowski–Bouligand dimension.
37
votes
Hausdorff dimension of R x X
Yes, it holds. In general it holds for any two separable metric spaces $X, Y$ with $Y$ totally bounded that
$$
\dim_H(X)+\dim_H(Y) \le \dim_H(X\times Y) \le \dim_H(X)+\dim_B(Y),
$$
where $\dim_B(Y)$ …
- 4,660
4
votes
Hausdorff dimension of a Cantor-like set
In Construction of 1-dimensional subsets of the reals not containing similar copies of given patterns Tamás Keleti constructs a compact set of Hausdorff dimension $1$ which contains no similar copy of …
- 4,660
1
vote
Packing measure and Kleinian groups
I tend to disagree with your thesis that Hausdorff measure and dimension is "more natural" than packing measure and dimension. It is true that Hausdorff dimension is far more widely used, but I attrib …
- 4,660
7
votes
Accepted
Geometric measures different from Hausdorff
Hausdorff, spherical Hausdorff and dyadic net measures not only give rise to the same dimension but, for a fixed value of $m$, are comparable up to constants that depend only on the ambient dimension …
- 4,660
1
vote
Isometrically-invariant measures and dilation of the Cantor set
This is too long for a comment. The following measure (defined on Borel sets) might be a counterexample to Question 1: let $\mathcal{I}_N$ be the collection of all left-closed, right-open intervals of …
- 4,660
13
votes
Accepted
The relation between Hausdorff dimension of an $n$-manifold and $n$
In a metrizable topological space, Hausdorff dimension is always larger or equal than the topological (covering) dimension. See Theorem 6.3.10 in Edgar's book "Measure, topology and fractal geometry". …
- 4,660
8
votes
Accepted
Can Hausdorff dimension make sets into a Tropical Semiring?
Actually, the inequality for the Hausdorff dimension of product sets goes the other way
(fixed in the OP now):
$$
\dim_{X\times Y}(U_1\times U_2) \ge \dim_X(U_1)+\dim_Y(U_2).
$$
And you probably ne …
- 4,660