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

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 …
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 …
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 …
Jörg Neunhäuserer's user avatar
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 …