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Questions about dimensions of possibly highly irregular or "rough" sets, Hausdorff–Besicovitch dimension and related concepts such as box-counting or Minkowski–Bouligand dimension.

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 …
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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 …
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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 …
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