Proper Holomorphic Embeddings of complements of large Cantor sets in $\Bbb C^2$

TL;DR

This paper constructs proper holomorphic embeddings of complements of large Cantor sets in 2 into 2, demonstrating embeddings for sets with Lebesgue measure arbitrarily close to 4.

Contribution

It provides a novel method to embed complements of large Cantor sets in 2 into 2, expanding understanding of complex embeddings of fractal sets.

Findings

Successfully embeds complements of large Cantor sets with measure close to 4

Demonstrates the flexibility of holomorphic embeddings for fractal complements

Advances techniques for embedding complex fractal structures

Abstract

We present a construction of a proper holomorphic embedding $f\colon \Bbb P^1\setminus C\hookrightarrow \Bbb C^2$, where C is a Cantor set obtained by removing smaller and smaller vertical and horizontal strips from a square of side 2, allowing to realize it to have Lebesgue measure arbitrarily close to 4.