  We prove a generalization of van der Corput's difference theorem for
sequences of vectors in a Hilbert space. This generalization is obtained by
establishing a connection between sequences of vectors in the first Hilbert
space with a vector in a new Hilbert space whose spectral type with respect to
a certain unitary operator is absolutely continuous with respect to the
Lebesgue measure. We use this generalization to obtain applications regarding
recurrence and multiple ergodic averages when we have measure preserving
automorphisms T and S that do not necessarily commute, but T has a maximal
spectral type that is mutually singular with the Lebesgue measure.
