# A removability theorem for Sobolev functions and detour sets

## TL;DR

This paper establishes conditions under which certain fractal-like sets, called detour sets, are removable for Sobolev functions, expanding understanding of the geometric properties influencing function extendability.

## Contribution

The paper introduces a new removability theorem for Sobolev functions applicable to detour sets with regular complementary components, including classical fractals.

## Key findings

- Detour sets with regular complements are W^{1,p}-removable for p > n.
- Examples include Sierpiński gasket, Apollonian gaskets, Julia sets.
- Theorem applies to a broad class of fractal sets with infinitely many components.

## Abstract

We study the removability of compact sets for continuous Sobolev functions. In particular, we focus on sets with infinitely many complementary components, called "detour sets", which resemble the Sierpi\'nski gasket. The main theorem is that if $K\subset \mathbb R^n$ is a detour set and its complementary components are sufficiently regular, then $K$ is $W^{1,p}$-removable for $p>n$. Several examples and constructions of sets where the theorem applies are given, including the Sierpi\'nski gasket, Apollonian gaskets, and Julia sets.