Integer-valued o-minimal functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916208608018432 |
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| author | Bhardwaj, Neer McCulloch, Raymond Ramachandran, Nandagopal Woo, Katharine |
| author_facet | Bhardwaj, Neer McCulloch, Raymond Ramachandran, Nandagopal Woo, Katharine |
| contents | We study $\mathbb{R}_{\textrm{an},\exp}$-definable functions $f:\mathbb{R}\to \mathbb{R}$ that take integer values at all sufficiently large positive integers. If $|f(x)|= O\big(2^{(1+10^{-5})x}\big)$, then we find polynomials $P_1, P_2$ such that $f(x)=P_1(x)+P_2(x)2^x$ for all sufficiently large $x$. Our result parallels classical theorems of Pólya and Selberg for entire functions and generalizes Wilkie's classification for the case of $|f(x)|= O(C^x)$, for some $C<2$.
Let $k\in \mathbb{N}$ and $γ_k=\sum_{j=1}^{k} 1/j$. Extending Wilkie's theorem in a separate direction, we show that if $f$ is $k$-$\textit{concordant}$ and $|f(x)|= O(C^{x})$, for some $C<e^{γ_k}+1$, then $f$ must eventually be given by a polynomial. This is an analog of a result by Pila for entire functions. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2404_10737 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Integer-valued o-minimal functions Bhardwaj, Neer McCulloch, Raymond Ramachandran, Nandagopal Woo, Katharine Logic Number Theory Primary 11U09, 03C64, Secondary 30D20, 26E05 We study $\mathbb{R}_{\textrm{an},\exp}$-definable functions $f:\mathbb{R}\to \mathbb{R}$ that take integer values at all sufficiently large positive integers. If $|f(x)|= O\big(2^{(1+10^{-5})x}\big)$, then we find polynomials $P_1, P_2$ such that $f(x)=P_1(x)+P_2(x)2^x$ for all sufficiently large $x$. Our result parallels classical theorems of Pólya and Selberg for entire functions and generalizes Wilkie's classification for the case of $|f(x)|= O(C^x)$, for some $C<2$. Let $k\in \mathbb{N}$ and $γ_k=\sum_{j=1}^{k} 1/j$. Extending Wilkie's theorem in a separate direction, we show that if $f$ is $k$-$\textit{concordant}$ and $|f(x)|= O(C^{x})$, for some $C<e^{γ_k}+1$, then $f$ must eventually be given by a polynomial. This is an analog of a result by Pila for entire functions. |
| title | Integer-valued o-minimal functions |
| topic | Logic Number Theory Primary 11U09, 03C64, Secondary 30D20, 26E05 |
| url | https://arxiv.org/abs/2404.10737 |