The Miles-Rubel-Taylor Theorem on Quotient Representations of Meromorphic Functions

Let f be a meromorphic function. In this chapter we describe the work of Joseph Miles, which completes the work in the last chapter concerning representations of f as the quotient of entire functions with small Nevan-linna characteristic. Miles showed that every set Z of finite λ-density is λ-balanceable. As a consequence of this and the work of Rubel and Taylor in the last chapter, there exist absolute constants A and B such that if f is any meromorphic function in the plane, then f can be expressed as f₁/f₂ where f₁ and f₂are entire functions such that T(r, f _i ) ≤ AT(Br, f) for i = 1, 2 and r > 0. It is implicit in the method of proof that for any B > 1 there is a corresponding A for which the desired representation holds for all f. Miles’ proof is ingenious, intricate, and deep. Miles also showed that, in general, B cannot be chosen to be 1 by giving an example of a meromorphic f such that if f = f₁/f₂, where f₁and f₂ are entire, then T(r, f₂) ≠ O(T(r, f)). We do not give this example here.