Integer Valued Definable Functions in $mathbb{R}_{an,exp}$

We give two variations on a result of Wilkies on unary functions defianble in $mathbb{R}_{an,exp}$ that take integer values at positive integers. Provided that the functions grows slower than the function $2^x$, 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 Wilkies result.