  We give two variations on a result of Wilkie's 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 Wilkie's result.
