Quasisymmetric extension on the real line

TL;DR

This paper characterizes subsets of the real line that allow quasisymmetric embeddings to be extended to the entire real line, providing a geometric criterion for such extensions.

Contribution

It offers a geometric characterization of sets in \\mathbb{R} that permit quasisymmetric embeddings to extend to the whole real line.

Findings

Identifies geometric conditions for extension of quasisymmetric embeddings.

Provides a criterion for when local embeddings extend globally.

Enhances understanding of quasisymmetric mappings on subsets of the real line.

Abstract

We give a geometric characterization of the sets $E\subset \mathbb{R}$ that satisfy the following property: every quasisymmetric embedding $f: E \to \mathbb{R}^n$ extends to a quasisymmetric embedding $f:\mathbb{R}\to\mathbb{R}^N$ for some $N\geq n$.