Let (X,B,μ,T)(X,B,μ,T) be a measure preserving system. We prove the pointwise convergence of the averages 1N2∑n,m=0N−1f1(Tnx)f2(Tmx)f3(Tn+mx)N21n,m=0∑N−1f1(Tnx)f2(Tmx)f3(Tn+mx) and of similar averages with seven bounded functions.
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