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In the previous lecture, we studied the recurrence properties of compact systems, which are systems in which all measurable functions exhibit almost periodicity – they almost return completely to themselves after repeated shifting. Now, we consider the opposite extreme of mixing systems – those in which all measurable functions (of mean zero) exhibit mixing – they become orthogonal to themselves after repeated shifting. (Actually, there are two different types of mixing, strong mixing and weak mixing, depending on whether the orthogonality occurs individually or on the average; it is the latter concept which is of more importance to the task of establishing the Furstenberg recurrence theorem.)

We shall see that for weakly mixing systems, averages such as $\frac{1}{N} \sum_{n=0}^{N-1} T^n f \ldots T^{(k-1)n} f$ can be computed very explicitly (in fact, this average converges to the constant $(\int_X f\ d\mu)^{k-1}$). More generally, we shall see that weakly mixing components of a system tend to average themselves out and thus become irrelevant when studying many types of ergodic averages. Our main tool here will be the humble Cauchy-Schwarz inequality, and in particular a certain consequence of it, known as the van der Corput lemma.

As one application of this theory, we will be able to establish Roth’s theorem (the k=3 case of Szemerédi’s theorem).