Multiple Polynomial Recurrence in Weyl Systems

In this work we give a full characterization of sets of multiple polynomial recurrence in Weyl systems, which are ergodic unipotent affine transformations on products of tori and finite abelian groups. In particular, we show that measurable and topological recurrence in Weyl systems coincide. Our analysis also yields a structure theorem for polynomial multicorrelation sequences in Weyl systems. These results stem from an in-depth study of the Weyl complexity of a set of polynomials and the introduction of a new concept: the $\mathit{\text{Weyl}}\mathit{\text{polynomials}}$ generated by a set of polynomials.