Integer-valued o-minimal functions

Abstract

We study $\mathbb{R}_{\textrm{an},\exp}$-definable functions $f:\mathbb{R}\to \mathbb{R}$ that take integer values at all sufficiently large positive integers. If $|f(x)|= O\big(2^{(1+10^{-5})x}\big)$, then we find polynomials $P_1, P_2$ such that $f(x)=P_1(x)+P_2(x)2^x$ 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 $|f(x)|= O(C^x)$, for some $C<2$. Let $k\in \mathbb{N}$ and $\gamma_k=\sum_{j=1}^{k} 1/j$. Extending Wilkie's theorem in a separate direction, we show that if $f$ is $k$-$\textit{concordant}$ and $|f(x)|= O(C^{x})$, for some $C<e^{\gamma_k}+1$, then $f$ must eventually be given by a polynomial. This is an analog of a result by Pila for entire functions.