Norm-variation of triple ergodic averages for commuting transformations

We prove an r -variation estimate, r > 4, in the norm for ergodic averages with respect to three commuting transformations.It is not known whether such estimates hold for all r 2 as in the analogous cases for one or two commuting transformations, or whether such estimates hold for any r < for more than three commuting transformations.1. Introduction 539 2. A collection of propositions on singular Brascamp-Lieb forms 545 3. Proof of Theorem 1.2 from Proposition 2.1 and Corollaries 2.2 and 2.4 553 4. Proof of Proposition 2.1 using Propositions 2.3 and 2.5 559 5. Proof of Proposition 2.3 using Propositions 2.5 and 2.6 561 6. Proof of Proposition 2.5 using Lemma 3 in Durcik and Thiele 2020 563 7. Proof of Proposition 2.6 using Propositions 2.7 and 2.8 569 8. Proof of Proposition 2.7 using Propositions 2.8, 2.9, and Durcik et al. 2022 576 9. Proof of Propositions 2.8 and 2.9