Integer Valued Definable Functions in $\mathbb{R}_{an,\exp}$

Gareth Jones; Shi Qiu

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arXiv · 2005.08852

Integer Valued Definable Functions in $\mathbb{R}_{an,\exp}$

Gareth Jones·Shi Qiu

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Abstract

We give two variations on a result of Wilkie's on unary functions defianble in $\mathbb{R}_{an,\exp}$ that take integer values at positive integers. Provided that the functions grows slower than the function $2^x$, Wilkie showed that is must be eventually equal to a polynomial. We show the same conclusion under a stronger growth condition but only assuming that the function takes values sufficiently close to a integers at positive integers. 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 primes), again under a stronger growth bound than that in Wilkie's result.

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Gareth Jones, Shi Qiu. 2020-05-18. Integer Valued Definable Functions in $\mathbb{R}_{an,\exp}$. https://arxiv.org/abs/2005.08852

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