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Avoiding the projective hierarchy in expansions of the real field by sequences

Author: Chris Miller
Journal: Proc. Amer. Math. Soc. 134 (2006), 1483-1493
MSC (2000): Primary 03C64; Secondary 26A12
Published electronically: October 5, 2005
MathSciNet review: 2199196
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Abstract: Some necessary conditions are given on infinitely oscillating real functions and infinite discrete sets of real numbers so that first-order expansions of the field of real numbers by such functions or sets do not define $\mathbb N$. In particular, let $f\colon\mathbb R\to\mathbb R$ be such that $\lim_{x\to+\infty}f(x)=+\infty$, $f(x)=O(e^{x^N})$ as $x\to +\infty$ for some $N\in\mathbb N$, $(\mathbb R, +,\cdot,f)$ is o-minimal, and the expansion of $(\mathbb R,+,\cdot)$ by the set $\{\,f(k):k\in\mathbb{N}\,\}$ does not define $\mathbb N$. Then there exist $r>0$ and $P\in\mathbb R[x]$ such that $f(x)=e^{P(x)}(1+O(e^{-rx}))$ as $x\to +\infty$.

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Chris Miller
Affiliation: Department of Mathematics, The Ohio State University, 231 West 18th Avenue, Columbus, Ohio 43210

Received by editor(s): February 4, 2004
Received by editor(s) in revised form: November 17, 2004
Published electronically: October 5, 2005
Additional Notes: This research was partially supported by NSF Grant No. DMS-9988855.
Communicated by: Carl G. Jockusch, Jr.
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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