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

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 $x\to +\infty$.