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Results tagged with hausdorff-dimension
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user 64462
Questions about dimensions of possibly highly irregular or "rough" sets, Hausdorff–Besicovitch dimension and related concepts such as box-counting or Minkowski–Bouligand dimension.
6
votes
1
answer
537
views
Direct proof that the set of badly approximable numbers have full Hausdorff dimension withou...
A badly approximable number is an $x$ for which there is a positive constant $c$ such that for all rational $p/q$ we have
$$\left|{ x - \frac{p}{q} }\right| > \frac{c}{q^2} \ . $$
The set of badly app …
- 1,565
1
vote
0
answers
98
views
Lower bound estimate for the sum $\sum \text{diam}(U)^d$ over all countable covers of a cube
This question is inspired from the definition of Hausdorff measure. Let $C$ be a closed unit hypercube in $\mathbb R^d$ (side length equal to one, including boundary. The cube itself is at top dimensi …
- 1,565
4
votes
1
answer
308
views
The Hausdorff codimension of singular matrices vs. the Hausdorff codimension of points with ...
Let $G:=SL(m+n,\mathbb R)$ and $\Gamma :=SL(m+n,\mathbb Z)$ and $X:=G/\Gamma$.
(1) Let $M$ denote the set of all $m \times n$ matrices with real entries. A matrix $A \in M$ is called $\textit{singular …
- 1,565