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 |
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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: | https://eprints.maths.manchester.ac.uk/id/eprint/1747 |

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