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Results tagged with dimension-theory
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user 23542
Hausdorff dimension, box dimension, packing dimension and similar concepts.
7
votes
2
answers
2k
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 self-simi …
- 1,648
4
votes
0
answers
119
views
A quantity that distinguishes finer than Hausdorff dimension
Consider sets $A\subseteq \mathbb{R}$ with Lebesgue measure zero and Hausdorff dimension one. For instance the set of real numbers with bounded entries in their continued fraction expansion have thi …
- 1,648
2
votes
Fractals of dimension zero
There are Moran geometric constructions which give closed generalized self-similar sets of Hausdorff-dimension zero. For the general construction see chapter 5 of Yakov B. Pesin Dimension Theory in D …
- 1,648
2
votes
0
answers
159
views
Dimension of Cartesian products
Is there a notion of dimension such that for all Borel sets $A,B\subseteq\mathbb{R}^{n}$ we have
$$ \dim(A\times B)=\dim(A)+\dim(B)?$$ For topological, Minkowski, packing and Hausdorff dimension this …
- 1,648