A generalization of van der Corput's difference theorem with applications to recurrence and multiple ergodic averages
arXiv:2303.11832 · doi:10.1080/14689367.2023.2230160
Abstract
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.
This was supposed to be version 5 of arXiv:2106.01123, but I accidentally created a brand new submission when trying to submit version 4. This final edition is the journal edition that includes some new applications and much better formatting. Dynamical Systems, 2023