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

TL;DR

This paper extends Wilkie's results on integer-valued definable functions in o-minimal structures, showing they are eventually polynomial under various growth and density conditions.

Contribution

It introduces new conditions under which integer-valued definable functions in 10_{an,0} are eventually polynomial, broadening Wilkie's original theorem.

Findings

Functions growing slower than 2^x are eventually polynomial.

Integer-valued functions close to integers at positive integers are polynomial.

Functions taking integer values on dense subsets like primes are polynomial.

Abstract

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.