Quasisymmetric embedding of the integer set and its quasiconformal extension

TL;DR

This paper establishes a precise condition under which a function from integers to reals can be extended to a quasiconformal map on the complex plane, linking quasisymmetry with quasiconformal extendability.

Contribution

It proves that an injection from integers to reals admits a quasiconformal extension iff it is quasisymmetric, providing a complete characterization.

Findings

Quasisymmetry characterizes extendability to quasiconformal maps.

The paper provides a necessary and sufficient condition for extension.

It bridges discrete and continuous geometric function theory.

Abstract

We prove that an injection from the integer set into the real line admits a quasiconformal extension to the complex plane if and only if it is quasisymmetric.