Integer-valued o-minimal functions

Abstract

We study ${\mathbb{R}}_{\text{an},\exp }$-definable functions $f:\mathbb{R}\to \mathbb{R}$ that take integer values at all sufficiently large positive integers. If $|f(x)|=O\left(2^{(1+10^{-5})x}\right)$, 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 Pólya 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 ${\hat{I}}_k^3={\sum }_{j=1}^{k}1/j$. Extending Wilkie's theorem in a separate direction, we show that if $f$ is $k$-$\text{concordant}$ and $|f(x)|=O(C^x)$, for some $C<e^{{\hat{I}}_k^3}+1$, then $f$ must eventually be given by a polynomial. This is an analog of a result by Pila for entire functions.

14 pages