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
N
\mathbb N
. In particular, let
f
:
R
→
R
f\colon \mathbb R\to \mathbb R
be such that
lim
x
→
+
∞
f
(
x
)
=
+
∞
\lim _{x\to +\infty }f(x)=+\infty
,
f
(
x
)
=
O
(
e
x
N
)
f(x)=O(e^{x^N})
as
x
→
+
∞
x\to +\infty
for some
N
∈
N
N\in \mathbb N
,
(
R
,
+
,
⋅
,
f
)
(\mathbb R, +,\cdot ,f)
is o-minimal, and the expansion of
(
R
,
+
,
⋅
)
(\mathbb R,+,\cdot )
by the set
{
f
(
k
)
:
k
∈
N
}
\{\,f(k):k\in \mathbb {N}\,\}
does not define
N
\mathbb N
. Then there exist
r
>
0
r>0
and
P
∈
R
[
x
]
P\in \mathbb R[x]
such that
f
(
x
)
=
e
P
(
x
)
(
1
+
O
(
e
−
r
x
)
)
f(x)=e^{P(x)}(1+O(e^{-rx}))
as
x
→
+
∞
x\to +\infty
.