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What is the Hausdorff dimension of the subset $S_c \subset [0,1]$ of points such that the critical exponent of their binary expansion is $c$? It's clear that $\dim_H S_{\infty}=1$, but what can be ...

Let $f:\mathbb R^n \to \mathbb R$ and $E := \{x \in X : f \text{ not Fréchet differentiable at }x\}$. Then $E$ is Borel measurable. It is well-known that Theorem If $f$ is convex, then the Hausdorff ...

Assume that $X$ is a compact metric space and $G$ is compact totally disconnected group. And $X$ has isometric free $G$-action i.e. $gx=x\Rightarrow g=e$. Then the following holds $${\rm dim}\ ...