Let N be a set of positive integers and let
\[
F
(
z
)
=
∑
A
n
z
n
F(z) = \sum {{A_n}{z^n}}
\]
be an entire function for which
A
n
=
0
(
n
∉
N
)
{A_n} = 0(n \notin N)
. It is reasonable to expect that, if D denotes the density of the set N in some sense, then
F
(
z
)
F(z)
will behave somewhat similarly in every angle of opening greater than
2
π
D
2\pi D
. For functions of finite order, the appropriate density seems to be the Pólya maximum density
P
\mathcal {P}
. In this paper we introduce a new density
D
\mathcal {D}
which is perhaps the appropriate density for the consideration of functions of unrestricted growth. It is shown that, if
|
I
|
>
2
π
D
|I| > 2\pi \mathcal {D}
, then
\[
log
M
(
r
)
∼
log
M
(
r
,
I
)
\log M(r) \sim \log M(r,I)
\]
outside a small exceptional set. Here
M
(
r
)
M(r)
denotes the maximum modulus of
F
(
z
)
F(z)
on the circle
|
z
|
=
r
|z| = r
and
M
(
r
,
I
)
M(r,I)
that of
F
(
r
e
i
θ
)
F(r{e^{i\theta }})
for values of
θ
\theta
in the closed interval I. The method used is closely connected with the question of approximating to functions on an interval by means of linear combinations of the exponentials
e
i
x
n
(
n
∈
N
)
{e^{ixn}}(n \in N)
.