Let A be a discrete subring of
C of rank 2. Let X be a compact subset of C with transfinite diameter not less
than unity or with transfinite diameter less than unity, void interior, and
connected complement. In an earlier paper we characterized the complex valued
functions on X which can be uniformly approximated by elements from the ring
of polynomials A[z]. In this paper the same problem is studied where X
is a compact subset of C with transfinite diameter d(X) less than unity
and with nonvoid interior. It is also studied for certain compact subsets of
Cn where n is any positive integer. These subsets will have the property
that every continuous function holomorphic on the interior is uniformly
approximable by complex polynomials. A large class of sets of this type is shown to
exist.
