We consider the class
B
\mathbf B
of entire functions of the form
\[
f
=
∑
p
j
exp
g
j
,
f=\sum p_j\exp g_j,
\]
where
p
j
p_j
are polynomials and
g
j
g_j
are entire functions. We prove that the zero-set of such an
f
f
, if infinite, cannot be contained in a ray. But for every region containing the positive ray there is an example of
f
∈
B
f\in \mathbf B
with infinite zero-set which is contained in this region.