Distortion of parallel lines under spatial quasiconformal mappings

Problem's Description

Problem. Is there a subset $S\subset \mathbb{R}^2$ such that $\dim_H S > 1$ and a quasiconformal map $f : \mathbb{R}^n\to\mathbb{R}^n, n\geq 3$, such that $f(\{x\}\times (0,1))$ contains no nontrivial rectifiable arc for any $x\in S$? \\

Remark. The planar version of this problem, i.e. for $n=2$ and $\dim_HS\leq1$, was stated in the paper by Balogh, Monti and Tyson, see Problem 6.4 in that paper. An affirmative answer was given by Bishop, Hakobyan and Williams, but the techniques used conformal mappings and cannot be generalized to higher dimensions. The present problem is a particular case of Question 9.4 in the paper of Bishop, Hakobyan and Williams.

1. Article Quasisymmetric dimension distortion of Ahlfors regular subsets of a metric space

   • Christopher J. Bishop,
   • Hrant Hakobyan,
   • Marshall Williams

   Geometric And Functional Analysis 26 (2), 379-421, 2016

2. Article Frequency of Sobolev and quasiconformal dimension distortion

   • Zoltan M. Balogh,
   • Roberto Monti,
   • Jeremy T Tyson

   Journal de Mathématiques Pures et Appliquées 99 (2), 125-149, 2013