We give two variations on a result of Wilkie’s [A. J. Wilkie, Complex continuations of ℝan,exp-definable unary functions with a diophantine application, J. Lond. Math. Soc. (2) 93(3) (2016) 547–566] on unary functions definable in ℝan,exp that take integer values at positive integers. Provided that the function grows slower (in a suitable sense) than the function 2x, 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.

中文翻译：

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ℝan,exp 中的整数值可定义函数

我们对 Wilkie 的 [AJ Wilkie, Complex continuations ofℝ一个,经验- 具有丢番图应用的可定义一元函数，J.伦敦。数学。社会党。(2)93(3) (2016) 547–566] 关于可在ℝ一个,经验取正整数的整数值。前提是函数的增长速度比函数慢（在适当的意义上）2X, Wilkie 证明它最终一定等于多项式。假设一个更强的增长条件，但仅假设函数在正整数处取值足够接近整数，我们表明该函数最终必须接近多项式。在一个不同的变体中，我们表明只要假设该函数在一个足够密集的正整数子集（例如素数）上取整数值就足够了，同样在比 Wilkie 的结果更强的增长限制下。