Pointwise convergence of averages along cubes

Abstract

Let $(X,\mathcal{B},\mu ,T)$ be a measure preserving system. We prove the pointwise convergence of the averages $$\frac{1}{N^2}\sum _{n,m=0}^{N-1}{f}_1(T^{n}x)f_2(T^{m}x)f_3(T^{n+m}x)$$ and of similar averages with seven bounded functions.

18 pages, latex, We have replaced Lemma 2 with a new one. We also have added a reference

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• Pointwise convergence along cubes for measure preserving systems