Integer-valued definable functions

Wilkie, A J and Jones, G O and Thomas, M E M (2012) Integer-valued definable functions. [MIMS Preprint]

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Abstract

We present a dichotomy, in terms of growth at infinity, of analytic functions definable in the real exponential field which take integer values at natural number inputs. Using a result concerning the density of rational points on curves definable in this structure, we show that if a function f : [0;1)n ! R is such that f(Nn) Z, then either supjxjr f(x) grows faster than exp(r), for some > 0, or f is a polynomial over Q.

Item Type: MIMS Preprint
Uncontrolled Keywords: O-minimality, number theory, CICADA
Subjects: MSC 2010, the AMS's Mathematics Subject Classification > 03 Mathematical logic and foundations
MSC 2010, the AMS's Mathematics Subject Classification > 11 Number theory
MSC 2010, the AMS's Mathematics Subject Classification > 26 Real functions
Depositing User: Prof Alex J Wilkie
Date Deposited: 05 Jan 2012
Last Modified: 20 Oct 2017 14:13
URI: http://eprints.maths.manchester.ac.uk/id/eprint/1747

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