A Cantor Set in the Unit Sphere in C~2 with Large Polynomial Hull

摘要

An old question of Walter Rudin, asked in connection with Banach algebras and approximation by polynomials, concerns how massive the polynomial hulls of Cantor sets may be. In contrast to expectation (note that Cantor sets have topo-logical dimension 0), it has been shown by Rudin, Vitushkin, and Henkin that the mentioned polynomial hull can be rather massive. Rudin himself constructed a Cantor set in C~2 whose polynomial hull (and even its rational hull) contains an analytic variety of dimension 1 [12, Thm. 5; 7, Thm. Ⅲ.2.5]. Later Vitushkin and Henkin gave examples of Cantor sets with interior points in the polynomial hull. The problem received new attention in connection with interest in topology on strictly pseudoconvex boundaries and hulls of their subsets, as well as in connection with removable singularities of CR functions. In particular, it was asked whether Cantor sets in the unit sphere in C~2 are polynomially convex. The expectation was that, for subsets of the sphere, the situation would change dramatically as for the case with some other problems. For example, totally real discs in C~2 are not necessarily polynomially convex, but if contained in the sphere they are so [9]. Further, the polynomial hull of a compact set in C~2 of finite 1-dimensional Hausdorff measure is not necessarily an analytic variety, but if the set is contained in the sphere it is so [14]. Moreover, the question about polynomial hulls of Cantor sets in the sphere has some relation 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.