Sobolev and quasiconformal distortion of intermediate dimension with applications to conformal dimension

TL;DR

This paper investigates how intermediate dimensions of sets are distorted under supercritical Sobolev and quasiconformal maps, extending classical theorems and providing new criteria for conformal dimension behavior with applications to fractal sets.

Contribution

It extends classical distortion theorems to intermediate dimensions and introduces new conditions for conformal dimension vanishing, with applications to fractal and product sets.

Findings

Extended Gehring–V"ais"al"a theorem to intermediate dimensions

Proved nonexistence of spaces with conformal dimension between 0 and 1

Provided new criteria for conformal box-counting dimension to vanish

Abstract

We study the distortion of intermediate dimension under supercritical Sobolev mappings and also under quasiconformal or quasisymmetric homeomorphisms. In particular, we extend to the setting of intermediate dimensions both the Gehring--V\"ais\"al\"a theorem on dilatation-dependent quasiconformal distortion of dimension and Kovalev's theorem on the nonexistence of metric spaces with conformal dimension strictly between zero and one. Applications include new contributions to the quasiconformal classification of Euclidean sets and a new sufficient condition for the vanishing of conformal box-counting dimension. We illustrate our conclusions with specific consequences for Bedford--McMullen carpets, samples of Mandelbrot percolation, and product sets containing a polynomially convergent sequence factor.