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Integer-valued definable functions in $ℝ_{an, exp}$

Gareth Jones

Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK

E-mail Address: gareth.jones-3@manchester.ac.uk

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and

Shi Qiu

Department of Mathematics, University of Manchester, Oxford Road, Manchester M13 9PL, UK

E-mail Address: shi.qiu-3@postgrad.manchester.ac.uk

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https://doi.org/10.1142/S1793042121500597Cited by:0 (Source: Crossref)

Abstract

We give two variations on a result of Wilkie’s [A. J. Wilkie, Complex continuations of $ℝ_{an, exp}$ -definable unary functions with a diophantine application, J. Lond. Math. Soc. (2) 93(3) (2016) 547–566] on unary functions definable in $ℝ_{an, exp}$ that take integer values at positive integers. Provided that the function grows slower (in a suitable sense) than the function $2^{x}$ , Wilkie showed that it must be eventually equal to a polynomial. Assuming a stronger growth condition, but only assuming that the function takes values sufficiently close to integers at positive integers, we show that the function must eventually be close to a polynomial. 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 the primes), again under a stronger growth bound than that in Wilkie’s result.

Keywords:

AMSC: 11U09, 03C64

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