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We prove an $r$-variation estimate, $r>4$, in the norm for ergodic averageswith respect to three commuting transformations. It is not known whether suchestimates hold for all $r\ge 2$ as in the analogous cases for one or twocommuting transformations, or whether such estimates hold for any $r<\infty$for more than three commuting transformations.

Norm-variation of triple ergodic averages for commuting transformations