Universal characteristic factors and Furstenberg averages

Let be an ergodic probability measure-preserving system. For a natural number we consider the averages

where , and are integers. A factor of is characteristic for averaging schemes of length (or -characteristic) if for any nonzero distinct integers , the limiting behavior of the averages in (*) is unaltered if we first project the functions onto the factor. A factor of is a -universal characteristic factor (-u.c.f.) if it is a -characteristic factor, and a factor of any -characteristic factor. We show that there exists a unique -u.c.f., and it has the structure of a -step nilsystem, more specifically an inverse limit of -step nilflows. Using this we show that the averages in (*) converge in . This provides an alternative proof to the one given by Host and Kra.