# 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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