On certain algebraic integers and approximation by rational functions with integral coefficients

David Geoffrey Cantor

Vol. 67 (1976), No. 2, 323–338

DOI: 10.2140/pjm.1976.67.323

Abstract

Let A be a finite set of integers {a₁,a₂,⋯,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 −a₁),1∕(𝜃 −a₂),⋯,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 − a₁),1∕(x − a₂),⋯ , 1∕(x − a_l), and if ∞∈ A,x itself?

Mathematical Subject Classification

Primary: 10F25, 10F25

Secondary: 12A15

Milestones

Received: 20 September 1974

Revised: 29 June 1976

Published: 1 December 1976