Caltech/UCLA Joint Analysis Seminar

Location: UCLA, MS 6221

A sequence of integers $\{n_k\} \subset \Z$ is said to be \emph{universally $L^p$-good} if for every measure-preserving system $(X,\mu,T)$ -- that is, every probability space $(X, \mu)$ equipped with a measure-preserving transformation, $T:X \to X$ -- and every $f \in L^p(X)$ the subsequence averages \[ A_N^{\{n_k\}} f := \frac{1}{N}\sum_{k=1}^N T^{n_k}f \] converge pointwise almost everywhere. In this language, Birkhoff's classical pointwise ergodic theorem states that the full sequence of integers is universally $L^1$-good. 

Obtaining pointwise convergence results for rougher, sparser sequences is much more challenging. 

For instance, (a special case of) Bourgain's celebrated Polynomial Ergodic Theorem states that the sequence $\{ n^2 \}$ is universally $L^p$-good for each $p>1$. Note that the squares are a zero-Banach-density subsequences of the integers; in fact, Bourgain used a probabilistic method to ``construct" \emph{extremely} sparse subsequences along which pointwise ergodic theorems held -- in the range $p >1$. Bourgain's approach, which relied on Fourier analysis, left the endpoint $p=1$ case unaddressed. Moreover, Bourgain's Polynomial Ergodic Theorem was recently shown to \emph{fail} at the endpoint case $p=1$; the distinction between pointwise ergodic theorems between $p >1$ and $p=1$ is fundamental. 

In 2009, LaVictoire succeeded in extending Bourgain's probabilistic construction to the $L^1$ setting: 

With $\{X_n\}$ a sequence of independent $\{0,1\}$ valued random variables (on a probability space $\Omega$) with expectations $n^{-a}$, $0< a < 1/2$, define the \emph{counting function} $a_n(\omega)$ to be the smallest integer subject to the constraint \[ X_1(\omega) + \dots + X_{a_n(\omega)}(\omega) = n.\] Then 

\begin{theorem}[LaVictoire's Random Ergodic Theorem] Almost surely, the sequence $\{a_n(\omega)\}$ is universally $L^1$-good. \end{theorem} 

We note that by the strong law of large numbers, such sequences have $a_{k}\approx k^{\frac{1}{1-a}}$ with full probability (i.e. almost surely there exist absolute constants $c_\omega < C_\omega$ so that $c_\omega k^{\frac{1}{1-a}} \leq a_k(\omega) \leq C_\omega k^{\frac{1}{1-a}}$). In other words, we have positive $L^1$-results for sequences almost as sparse as the squares. 

The first aim of this talk will be to sketch LaVictoire's random ergodic theorem; we will then discuss a \emph{Wiener-Wintner} variant of his result obtained in collaboration with Pavel Zorin-Kranich (University of Bonn): 

\begin{theorem} In the above setting, let $0 < \epsilon < 1$ be arbitrary; almost surely, the following holds: 

For each measure-preserving system $(X,\mu,T)$ and each $f \in L^1(X)$ the averages \[ \frac{1}{N} \sum_{n=1}^N e( n^{1+ \epsilon} ) T^{a_n(\omega)}f \] converge to zero pointwise $\mu$-almost everywhere. \end{theorem}