Rationally convex sets on the unit sphere in C2

Let X be a rationally convex compact subset of the unit sphere S in C-2, of three-dimensional measure zero. Denote by R(X) the uniform closure on X of the space of functions P/Q, where P and Q are polynomials and Q not equal 0 on X. When does R(X)=C(X)? Our work makes use of the kernel function for the (delta) over bar (b) operator on S, introduced by Henkin in [5] and builds on results obtained in Anderson-Izzo-Wermer [3]. We de. ne a real-valued function epsilon(X) on the open unit ball int B, with epsilon(X)(z, w) tending to 0 as (z, w) tends to X. We give a growth condition on epsilon(X)(z, w) as (z, w) approaches X, and show that this condition is sufficient for R(X)=C(X) (Theorem 1.1). In Section 4, we consider a class of sets X which are limits of a family of Levi-flat hypersurfaces in int B. For each compact set Y in C-2, we denote the rationally convex hull of Y by (Y) over cap Y. A general reference is Rudin [8] or Aleksandrov [1].