Uniformization of Cantor sets with bounded geometry

TL;DR

This paper extends a known result by showing that uniformly perfect and disconnected sets in higher dimensions can be quasisymmetrically transformed into standard Cantor sets, generalizing MacManus's 2D result.

Contribution

It generalizes MacManus's 2D result to higher dimensions, establishing a quasisymmetric equivalence between certain fractal sets and standard Cantor sets in n+1 dimensions.

Findings

A compact subset of n is uniformly perfect and disconnected iff it is quasiconformal to the standard Cantor set.

Provides a quasisymmetric taming of such sets in higher dimensions.

Generalizes known 2D results to higher-dimensional Euclidean spaces.

Abstract

In this note we provide a quasisymmetric taming of uniformly perfect and uniformly disconnected sets that generalizes a result of MacManus from 2 to higher dimensions. In particular, we show that a compact subset of $\mathbb{R}^n$ is uniformly perfect and uniformly disconnected if and only if it is ambiently quasiconformal to the standard Cantor set $\mathcal{C}$ in $\mathbb{R}^{n+1}$.