For a function
f
f
analytic in the unit disc
D
D
, and for each
λ
>
0
\lambda > 0
, let
L
(
λ
)
=
{
z
∈
D
:
|
f
(
z
)
|
=
λ
}
L\left ( \lambda \right ) = \left \{ {z \in D:\left | {f\left ( z \right )} \right | = \lambda } \right \}
denote a level set for
f
f
. We introduce a class
L
\mathcal {L}
, of functions characterized by geometric properties of a collection of sets
{
L
(
λ
n
)
}
\left \{ {L\left ( {{\lambda _n}} \right )} \right \}
, where
{
λ
n
}
\left \{ {{\lambda _n}} \right \}
is an unbounded sequence. We show that
L
1
{\mathcal {L}_1}
, is a proper subclass of the class
L
\mathcal {L}
of G. R. MacLane. Let
A
∞
{A_\infty }
denote the set of points
e
i
θ
{e^{i\theta }}
at which the function
f
f
has
∞
\infty
as an asymptotic value, and let
F
(
f
)
F\left ( f \right )
denote the set of Fatou points of
f
f
. We prove that for a function
f
f
in the class
L
1
{\mathcal {L}_1}
, if
Γ
\Gamma
is an arc of the unit circle such that
Γ
∩
A
∞
=
∅
\Gamma \cap {A_\infty } = \emptyset
, then almost every point of
Γ
\Gamma
belongs to
F
(
f
)
F\left ( f \right )
.