 
Summary: November 16, 2003
POINTWISE CONVERGENCE ALONG CUBES FOR MEASURE
PRESERVING SYSTEMS
BY I. ASSANI
Abstract. Let (X, B, µ) be a probability measure space and T1, T2 , T3 three not neces
sarily commuting measure preserving transformations on (X, B, µ). We prove that for all
bounded functions f1, f2, f3 the averages
1
N2
N
n,m=1
f1(Tn
1 x)f2(Tm
2 x)f3(Tn+m
3 x)
converges a.e.. Generalizations to averages of 2k
 1 functions are also given for not
necessarily commuting weakly mixing systems.
1. Introduction
In [A1] and [A2] we proved that if T is a measure preserving transformation on (X, B, µ)
