Gaussian integer points of analytic functions in a half-plane

Abstract

AbstractA classical result of Pólya states that 2zis the slowest growing transcendental entire function taking integer values on the non-negative integers. Langley generalised this result to show that 2zis the slowest growing transcendental function in the closed right half-plane Ω = {z∈: Re(z) ≥ 0} taking integer values on the non-negative integers. LetEbe a subset of the Gaussian integers in the open right half-plane with positive lower density and letfbe an analytic function in Ω taking values in the Gaussian integers onE. Then in this paper we prove that iffdoes not grow too rapidly, thenfmust be a polynomial. More precisely, there existsL> 0 such that if either the order of growth offis less than 2 or the order of growth is 2 and the type is less thanL, thenfis a polynomial.