Entire functions having a concordant value sequence

A classic theorem of Pólya shows that 2 ^z is, in a strong sense, the “smallest” transcendental entire function that is integer valued on ℕ. An analogous result of Gel’fond concerns entire functions that are integer valued on the setX _a={a ⁿ:n ∈ ℕ}, wherea ∈ ℕ,|a|≥ 2. LetX=ℕ orX=X _a andκ ∈ ℕ orκ=∞. This paper pursues analogous results for entire functionsf having the following property: on any finite subsetD ofX with#D≤κ+1, the valuesf(z),z ∈D admit interpolation by an element of ℤz]. The results obtained assert that if the growth off is suitably restricted then the restriction off toX must be a polynomial. WhenX=X _a andκ<∞ a “smallest” transcendental entire function having the requisite property is constructed.