For
x
,
y
∈
R
x,y \in \mathbb {R}
and
f
∈
L
2
(
R
)
f \in L^2(\mathbb {R})
, define
(
x
,
y
)
f
(
t
)
=
e
2
π
i
y
t
f
(
t
+
x
)
(x,y) f(t) = e^{2\pi iyt} f(t+x)
and if
Λ
⊆
R
2
\Lambda \subseteq \mathbb {R}^2
, define
S
(
f
,
Λ
)
=
{
(
x
,
y
)
f
∣
(
x
,
y
)
∈
Λ
}
S(f, \Lambda ) = \{(x,y)f \mid (x,y) \in \Lambda \}
. It has been conjectured that if
f
≠
0
f\ne 0
, then
S
(
f
,
Λ
)
S(f,\Lambda )
is linearly independent over
C
\mathbb {C}
; one motivation for this problem comes from Gabor analysis. We shall prove that
S
(
f
,
Λ
)
S(f, \Lambda )
is linearly independent if
f
≠
0
f \ne 0
and
Λ
\Lambda
is contained in a discrete subgroup of
R
2
\mathbb {R}^2
, and as a byproduct we shall obtain some results on the group von Neumann algebra generated by the operators
{
(
x
,
y
)
∣
(
x
,
y
)
∈
Λ
}
\{(x,y) \mid (x,y) \in \Lambda \}
. Also, we shall prove these results for the obvious generalization to
R
n
\mathbb {R}^n
.