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Integer-valued definable functions in \(\mathbb{R}_{\mathrm{an} , \exp} \). (English) Zbl 1501.11110

A classical theorem of Pólya states that if \(f\) is an entire function taking integer values at the non-negative integers and satisfying that \(f(z) = O(|z|^M 2^{|z|})\) as \(z \to \infty\), for some \(M > 0\), then there exist polynomials \(g_1, g_2\) with \(f(z) \equiv g_1(z)2^z + g_2(z)\). In particular if \(\lim\sup_{r\to\infty} O(|z|^M 2^{|z|})/2^r<1\), then \(f\) is a polynomial. In [G. O. Jones et al., Bull. Lond. Math. Soc. 44, No. 6, 1285–1291 (2012; Zbl 1275.03130)], the authors showed analogous results for functions \(f \colon (0,\infty) \to \mathbb{R}\) definable in certain o-minimal structures. In the present paper, the authors considered two variants of the same problem: the first one replaces the integer-valued assumption by supposing that \(f\) takes values sufficiently close to integers; the second variant weakens the same assumption by supposing that \(f\) is integer-valued on a sufficiently dense subset of the positive integers. Formally, their results correspond to the following two theorems:
1.
There is a \(\delta>0\) with the following property. Suppose that \(f\colon [0,\infty)\to \mathbb{R}\) is definable and analytic, and that there exists \(c_0>0\) such that for all positive integers \(n\) there is an integer \(m_n\) such that \[ |f(n)-m_n|<c_0 e^{-3n}. \] If there are \(c_1>0\) and \(\delta'<\delta\) such that \(|f(x)|<c_1e^{\delta' x}\) then there is a polynomial \(Q\) such that \( Q(n)=m_n\) for all sufficiently large \(n\).
2.
Fix a subset \(A\subseteq \mathbb{N}\) such that there is a positive real \(\lambda\) such that for all sufficiently large \(T\) we have \[ \frac{T}{(\log T)^\lambda} \ll \# A \cap [0,T] \ll \frac{T}{(\log T)^\lambda}. \] Suppose that \(f\colon [0,\infty)\to \mathbb{R}\) is definable and analytic, and such that \(f(n)\) is an integer for \(n\in A\). If there exist \(\alpha>0\) and \(c_1>0\) such that \[ |f(x)|<c_1 \exp \left(\frac{x}{(\log x)^{2\lambda+2+\alpha}}\right) \] then \(f\) is a polynomial.
The second results shows, for example, that if \(|f(x)|<c \exp\left(\frac{x}{(\log x)^5}\right)\) and \(f\) is integer-valued on the primes, then \(f\) is a polynomial.

MSC:

11U09 Model theory (number-theoretic aspects)
03C64 Model theory of ordered structures; o-minimality

Citations:

Zbl 1275.03130
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References:

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