Integer-valued definable functions

Abstract

We study integer-valued functions definable in $\mathbb{R}_{\text{an},\exp}$. We first give several variations on a result of Wilkie's, and show that, under certain growth conditions, unary functions definable in $\R_{\text{an},\exp}$ that take integer values at some sufficiently dense subset of positive integers must be polynomials. We then study functions that take values sufficiently close to integers at positive integers. Under certain growth conditions, we show that such functions must be close to a polynomial. The methods here combine Wilkie's results on continuation with transcendence methods. We then consider various results of P\'{o}lya-type for definable functions of several variable. Finally we use Wilkie's methods to check that some of his results on definable continuation go through in certain reducts of $\mathbb{R}_{\text{an},\exp}$, namely expansions of the real field by certain Weierstrass systems and the exponential function.

Date of Award	1 Aug 2021
Original language	English
Awarding Institution	The University of Manchester
Supervisor	Marcus Tressl (Supervisor) & Gareth Jones (Supervisor)