Integer-valued o-minimal functions

We study Ran,exp-definable functions f:R→ R that take integer values at all sufficiently large positive integers. If |f(x)|= O(2^(1+10⁻⁵)x), then we find polynomials P₁, P₂ such that f(x)=P₁(x)+P₂(x)2ˣ for all sufficiently large x. Our result parallels classical theorems of P\'olya and Selberg for entire functions and generalizes Wilkie's classification for the case of |f(x)|= O(Cˣ), for some $C<2$. Let k∈ N and γₖ=∑ⱼ₌₁ᵏ 1/j. Extending Wilkie's theorem in a separate direction, we show that if f is k-concordant and |f(x)|= O(Cˣ), for some C<eγₖ+1, then f must eventually be given by a polynomial. This is an analog of a result by Pila for entire functions.