A Cantor set whose polynomial hull contains no analytic discs

TL;DR

This paper constructs a specific Cantor set in complex three-dimensional space whose polynomial hull is larger than the set itself but contains no analytic discs, extending previous results on polynomial hulls.

Contribution

It generalizes Wermer's result by providing a new example of a Cantor set with a polynomial hull that lacks analytic discs.

Findings

Existence of a Cantor set in ${f C}^3$ with a hull containing no analytic discs.

The polynomial hull of the constructed set is strictly larger than the set itself.

The result extends the understanding of polynomial hulls without analytic discs.

Abstract

A generalization of a result of Wermer concerning the existence of polynomial hulls without analytic discs is presented. As a consequence it is shown that there exists a Cantor set $X$ in ${\mathbb C}^3$ whose polynomial hull is strictly larger than $X$ but contains no analytic discs.