Throughout this paper A will
denote a discrete subring of the complex number plane C with rank 2. For
example, A could be the Gaussian integers Z + iZ, where Z denotes the
rational integers, or the ring of integers of any imaginary quadratic field. We
are concerned with characterizing those functions defined on a compact
subset X of C which can be uniformly approximated by polynomials with
coefficients in A. We say that such functions are A-approximable on X. We also
consider the real case where X is any compact subset of the reals R and
the coefficients of the approximating polynomials lie in Z or any discrete
subring of R. The real case is completely solved in the sense that a necessary
and sufficient condition in order that a function can be so approximated
is found. The complex case is solved if, in addition to being compact, X
either has transfinite diameter at least unity or void interior and connected
complement.
