A Cantor set in the unit sphere in $\mathbb{C}^2$ with large polynomial hull

Abstract

It is an old question how massive polynomial hulls of Cantor sets in $\mathbb{C}^n$ can be. In contrast to expectation e.g. Rudin, Vitushkin and Henkin showed on examples that it can be rather massive. Motivated by problems of holomorphic convexity of subsets of strictly pseudoconvex boundaries and removable singularities the question was asked for Cantor sets in the unit sphere. It was known that tame Cantor sets in the unit sphere are polynomially convex. We give an example of a wild Cantor set in the sphere whose polynomial hull contains a large ball. In some sense this can be opposed to a still open conjecture of Vitushkin on the existence of a lower bound for the diameter of the largest boundary component of a relatively closed complex curve in the ball passing through the origin.