Norm convergence of multiple ergodic averages for commuting transformations

Let T 1 ,…, T l : X → X be commuting measure-preserving transformations on a probability space (X, X, μ ) . We show that the multiple ergodic averages 1N ∑ ₙ₌₀N-1 f₁(T₁ⁿ x) ⋯ fₗ(Tₗⁿ x) are convergent in L²(X,X,μ ) as N → ∞ for all f₁,… ,fₗ ∈ L^∞ (X,X,μ ) ; this was previously established for l =2 by Conze and Lesigne [J. P. Conze and E. Lesigne. Théorèmes ergodique por les mesures diagonales. Bull. Soc. Math. France 112 (1984), 143–175] and for general l assuming some additional ergodicity hypotheses on the maps T i and T i T j −1 by Frantzikinakis and Kra [N. Frantzikinakis and B. Kra. Convergence of multiple ergodic averages for some commuting transformations. Ergod. Th. & Dynam. Sys. 25 (2005), 799–809] (with the l =3 case of this result established earlier by Zhang [Q. Zhang. On the convergence of the averages 1N ∑ ₙ₌₁N f₁(Rⁿ x) f₂(Sⁿ x) f₃(Tⁿ x) . Mh. Math. 122 (1996), 275–300]). Our approach is combinatorial and finitary in nature, inspired by recent developments regarding the hypergraph regularity and removal lemmas, although we will not need the full strength of those lemmas. In particular, the l =2 case of our arguments is a finitary analogue of those by Conze and Lesigne.