# Nilpotent groups and non-conventional ergodic theorems

## Hillel Furstenberg (Jerusalem)

### Abstract

The Erdos-Turan conjecture on the existence of arbitrarily long
arithmetic progressions in subsets of integers of positive density was
proved by Szemeredi using combinatorial methods. However it is now
understood to be related to deep recurrence properties of dynamical
systems. An alternate way of establishing these recurrence results is by
means of "non-conventional" ergodic theorems relating to averages such
as

$$\frac{1}{N} \sum_{n=1}^N f(T^{n}x) g(T^{2n}x) ... h(T^{kn}x)$$
The study of these averages is of independent
interest as it leads to a new type of structural analysis of
ergodic systems. In this structural analysis a special role is
played by nilpotent groups and their homogeneous spaces. We will
try to explain these various connections, particularly attempting
to clarify the special role of nilpotent groups.

[Newton Institute]
[Seminars on the Web]
[Monday Seminars]
[Furstenberg, 2000-02-14]
abstract.html

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