Pleasant extensions retaining algebraic structure, II

In this paper, we combine the general tools developed in [5] with several ideas taken from earlier work on one-dimensional nonconventional ergodic averages by Furstenberg and Weiss [17], Host and Kra [21] and Ziegler [44] to study the averages (Formula presented.) associated to a triple of directions p₁, p₂, p₃ ∈ ℤ² that lie in general position along with 0 ∈ ℤ². We show how to construct a “pleasant” extension of an initiallygiven ℤ²-system for which these averages admit characteristic factors with a very concrete description, involving the same structure as for those in [2] together with two-step pro-nilsystems (reminiscent of [21] and its predecessors). We also use this analysis to construct pleasant extensions and then prove norm convergence for the polynomial nonconventional ergodic averages (Formula presented.) associated to two commuting transformations T₁, T₂.