Von Neumann algebras and linear independence of translates

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Abstract
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 <inline-formula...

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