## 2012.1: Integer-valued definable functions

2012.1:
A J Wilkie, G O Jones and M E M Thomas
(2012)
*Integer-valued definable functions.*

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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 2000 > 03 Mathematical logic and foundations MSC 2000 > 11 Number theory MSC 2000 > 26 Real functions |

MIMS number: | 2012.1 |

Deposited By: | Prof Alex J Wilkie |

Deposited On: | 05 January 2012 |

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