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

Authors:	Chris Miller

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

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 and $P\in\mathbb Rx]$ such that $f(x)=e^{P(x)}(1+O(e^{-rx}))$ as $x\to +\infty$ .

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