Search results for "Metric geometry"
showing 2 items of 222 documents
Quasiconformal Jordan Domains
2020
We extend the classical Carath\'eodory extension theorem to quasiconformal Jordan domains $( Y, d_{Y} )$. We say that a metric space $( Y, d_{Y} )$ is a quasiconformal Jordan domain if the completion $\overline{Y}$ of $( Y, d_{Y} )$ has finite Hausdorff $2$-measure, the boundary $\partial Y = \overline{Y} \setminus Y$ is homeomorphic to $\mathbb{S}^{1}$, and there exists a homeomorphism $\phi \colon \mathbb{D} \rightarrow ( Y, d_{Y} )$ that is quasiconformal in the geometric sense. We show that $\phi$ has a continuous, monotone, and surjective extension $\Phi \colon \overline{ \mathbb{D} } \rightarrow \overline{ Y }$. This result is best possible in this generality. In addition, we find a n…
On arithmetic sums of Ahlfors-regular sets
2021
Let $A,B \subset \mathbb{R}$ be closed Ahlfors-regular sets with dimensions $\dim_{\mathrm{H}} A =: \alpha$ and $\dim_{\mathrm{H}} B =: \beta$. I prove that $$\dim_{\mathrm{H}} [A + \theta B] \geq \alpha + \beta \cdot \tfrac{1 - \alpha}{2 - \alpha}$$ for all $\theta \in \mathbb{R} \, \setminus \, E$, where $\dim_{\mathrm{H}} E = 0$.