Uniform Wiener-Wintner Theorem - proof

Question

I am looking for proof of uniform version of Wiener-Wintner theorem:

Let $(X, \mathcal{A}, \mu, T)$ be an ergodic measure preserving system. For $f \in L^1(\mu)$ which is orthogonal to the Kronecker Factor of $(X, \mathcal{A}, \mu, T)$ (i.e. orthogonal to all eigenfunctions of $T$ acting on $L^2(\mu)$) we have convergence

$\underset{N \rightarrow \infty}{lim}\ \underset{|z|=1}{sup}\ | \frac{1}{N} \sum_{n=0}^{N-1}z^n f(T^n x)| = 0$ for a.e. $x \in X$.

($z$ are taken from unit circle on complex plane)

I know that the proof can be found in articles

1. I. Assani, A Wiener-Wintner property for the helical transform, Ergodic Theory and Dynamical Systems 12 (1992), 185-194.

(which is avaible on Cambridge Journals, but I don't have free access to their library)

2. E. Lesigne, Spectre quasi-discret et thdorbme ergodique de Wiener-Wintner pour les polynomes, Ergodic Theory and Dynamical Systems .

(I am unable to find that article and I also don't speak French)

and book

3. I. Assani. Wiener Wintner Ergodic Theorems. World Science Pub Co Inc, May 2003

(Which is unavaible in electronic version)

This theorem is also mentionend (probably for the first time) at the beginning of

4. J. Bourgain, Double recurrence and almost sure convergence, Journal fiir die reine und angewandte Mathematik 404 (1990), 140-161,

but it seems to lack proof. Do you know any other article or book where I can find proof of this theorem?

Answer 1

There are two results called the uniform wiener-wintner theorem, and you're refering to the stronger one (taking a limsup(, which is indeed proven by Bourgain.

Bourgain documented his techniques for obtaining convergence in several articles and exposes (check the Israel journal where he published most of his work, he also have an IHES paper, and the expose appears in GAFA notes and there's a paper in french by Host as Bourbaki report).

Anyways, you can recover the result (with the proof which relays on the van-der-corput trick) in the Host-Kra article, Prop 2.9 - http://www.math.northwestern.edu/~kra/papers/uniformity.pdf As you might not be familiar with the terminology, the $2$-nd Gowers norm is simply the following term - $\frac{1}{N^2}\sum_{n,h=0}^{N-1}f(T^{n}x)f(T^{n+h}x)$, maybe you should conjugate one term, but it doesn't really matter. The convergence of those averages (for functions which are orthogonal to the Kronecker factor) is by the mixing property.

The weak uniform wiener-wintner is due to Bellow and Losert, their article is available online here, Cor 2.7 - http://www.ams.org/journals/tran/1985-288-01/S0002-9947-1985-0773063-8/S0002-9947-1985-0773063-8.pdf The mixing enters here as the associated spectral measure (of the system modulo the Kronecker factor) is atom-less.