 
Summary: POINTWISE CONVERGENCE OF AVERAGES ALONG CUBES II
I. ASSANI
Abstract. Let (X, B, µ, T) be a measure preserving system. We prove the pointwise
convergence of averages along cubes of 2k
 1 bounded and measurable functions for all k.
1. Introduction
Let (X, B, µ, T) be a dynamical system where T is a measure preserving transformation on
the measure space (X, B, µ, T). In [1] we proved the pointwise convergence of the averages
1
N2
N1
n,m=0
f1(Tn
x)f2(Tm
x)f3(Tn+m
x)
and of similar averages with seven bounded functions fi. We also showed that if T is weakly
mixing then similar averages for 2k 1 bounded functions converge a.e to the product of the
integrals of the functions fi. The averages of three functions were used in [3] to generalize
Khintchine recurrence result [5]. In [2] B.Host and B.Kra proved that the averages of 2k 1
