Quasiconformal mappings that highly distort dimensions of many parallel lines

TL;DR

This paper constructs a quasiconformal mapping in n-dimensional space that significantly distorts the Hausdorff dimension of many parallel lines, addressing a previously open question in geometric analysis.

Contribution

It provides a novel construction of quasiconformal maps that can highly distort dimensions of multiple parallel lines simultaneously.

Findings

Successfully distorts the Hausdorff dimension of many parallel lines

Answers an open question by Balogh, Monti, and Tyson

Demonstrates the extent of dimension distortion achievable by quasiconformal maps

Abstract

We construct a quasiconformal mapping of $n$-dimensional Euclidean space, $n \geq 2$, that simultaneously distorts the Hausdorff dimension of a nearly maximal collection of parallel lines by a given amount. This answers a question of Balogh, Monti, and Tyson.