Integer Valued Definable Functions in Ran exp

We give two variations on a result of Wilkie's on unary functions defianble in Ran,exp that take integer values at positive integers. Provided that the functions grows slower than the function 2x, Wilkie showed that is must be eventually equal to a polynomial. We show the same conclusion under a stronger growth condition but only assuming that the function takes values sufficiently close to a integers at positive integers. In a different variation we show that it suffices to assume that the function takes integer values on a sufficiently dense subset of the positive integers(for instance primes), again under a stronger growth bound than that in Wilkie's result.