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Pleasant extensions retaining algebraic structure, II

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Abstract

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

$$\frac{1}{N}\sum\limits_{n = 1}^N {({f_1} \circ {T^{n{p_1}}}} )({f_2} \circ {T^{n{p_2}}})({f_3} \circ {T^{n{p_3}}}){\text{ }}{f_1},{f_2},{f_3} \in {L^\infty }(\mu )$$

associated to a triple of directions p 1, p 2, p 3 ∈ 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

$$\frac{1}{N}\sum\limits_{n = 1}^N {({f_1} \circ T_1^{{n^2}}} )({f_2} \circ T_1^{{n^2}}T_2^n)$$

associated to two commuting transformations T 1, T 2.

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  1. Courant Institute, New York University, New York, NY, 10012, USA

    Tim Austin

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Austin, T. Pleasant extensions retaining algebraic structure, II. JAMA 126, 1–111 (2015). https://doi.org/10.1007/s11854-015-0013-5

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  • DOI: https://doi.org/10.1007/s11854-015-0013-5

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