Let A be a finite set of integers
{a_{1},a_{2},⋯,a_{l}} and (possibly) ∞. Let X be a nonempty closed subset of C ∪{∞}, the
field of complex numbers together with ∞, under the topology of the Riemann
sphere. Suppose that X is symmetric with respect to the field of real numbers R
(i.e. if z ∈ X then z ∈ X) and disjoint from A. We are interested in the following two
problems:
I. Under what conditions do there exist, for each neighborhood N of X, infinitely
many algebraic numbers 𝜃 such that 1∕(𝜃 −a_{1}),1∕(𝜃 −a_{2}),⋯,1∕(𝜃 −a_{l}) are algebraic
integers and, if ∞∈ A, 𝜃 is itself an algebraic integer, such that all of the (algebraic)
conjugates of 𝜃 lie in N?
II. If X has empty interior and connected complement, then the polynomials are
dense in the ring of continuous functions of X. What is the uniform closure of the
polynomials with integral coefficients in 1∕(x − a_{1}),1∕(x − a_{2}),⋯ , 1∕(x − a_{l}), and if
∞∈ A,x itself?
