Every Planar Set has a Conformally Removable Subset with the Same Hausdorff Dimension

TL;DR

This paper proves that for any compact set in the complex plane, there exists a conformally removable subset sharing the same Hausdorff dimension, advancing understanding of conformal removability and fractal geometry.

Contribution

It establishes that every planar set contains a conformally removable subset with identical Hausdorff dimension, a novel result linking fractal geometry and conformal removability.

Findings

Existence of conformally removable subsets with same Hausdorff dimension as any given compact set.

Extension of conformal removability concepts to arbitrary planar sets.

New connections between Hausdorff dimension and conformal removability.

Abstract

In this paper we show that given any compact set $E \subset \hat{\mathbb{C}}$, we can always find a conformally removable subset with the same Hausdorff dimension as $E$.