This post is nothing more than a complement of ERT12. I preferred to separate it to show its own importance, specially in the historic development of the achievements. With the help of the tools developed in ERT12, we prove the
His proof relied on a Fourier-analytic argument of energy increment for functions: one decomposes a function as , where is good and is bad in a specific sense. If the effect of is large, it is possible to break it into good and bad parts again and so on. In each step, the “energy” of increases a fixed amount. Being bounded, it must stop after a finite number of steps. At the end, controls the behavior of and for it the result is straightforward. This follows the same philosophy of Calderón-Zygmund theory of singular integrals in harmonic analysis. See this article of Alexander Arbieto, Carlos Matheus and Carlos Gustavo Moreira for further details.
This represented the first affirmative result in the direction of Szemerédi theorem, at that time known as the Erdös-Turán conjecture. Only 16 year later, in 1969, Szemerédi proved the existence of arithmetic progressions of length 4 and, finally, in 1975 proved the conjecture in its full generality.
By Furstenberg correspondence principle (see ERT4), Theorem 1 will follow from
In order to prove this, we apply Koopman-von Neumann theorem obtained in ERT12 to write
where the other terms are the sum of 7 expressions of the type
where at least one of is weak mixing. It turns out that the appearance of at least one weak mixing function vanishes the above limit. This is the content of the
We can also assume this because
and so the positions of can be switched. To prove (2), we make use of van der Corput trick. Defining , we have
where in the last equality we used the ergodicity of . If ,
which guarantees that
Proposition 3 proves Theorem 2. In fact, by (1),
The component is non-negative, by Proposition 4 of ERT1. In addition,
because , being weak mixing, is orthogonal to the constant functions, that is, has zero mean. This guarantees the non-negativity of the inferior limit. To prove it is positive, we proceed as in ERT10: given , the norm for a syndetic set of iterates . If is sufficiently small, to each of these iterates the integral is positive. This concludes the proof of Theorem 2.