Multiple ergodic averages — Problem 32

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

A system is a probability space $(X,\mathcal X,\mu)$ with invertible measure-preserving transformations; multiple transformations are assumed to commute. Write $T^j f=f\circ T^j$, and $T^j A$ for the image of a measurable set under $T^j$. A system with one transformation is ergodic if every invariant measurable set has measure zero or one. The space $L^\infty(\mu)$ consists of essentially bounded complex measurable functions. Let $(\sigma_j)_{j\ge1}$ be a nonincreasing sequence in $(0,1]$ with $j\sigma_j\to\infty$. On an auxiliary probability space $(\Omega,\mathcal F,\mathbb P)$ take independent Bernoulli variables $X_j$ with $\mathbb P(X_j=1)=\sigma_j$. Enumerate the selected positive integers increasingly: $a_n(\omega)=\min\{k:\sum_{j=1}^kX_j(\omega)=n\}$. This sequence is infinite almost surely. For a transformation $T$, let $\mathcal I_T$ be its invariant $\sigma$-algebra. Conditional expectation $\mathbb E(f\mid\mathcal I_T)$ is the orthogonal projection in $L^2$ onto the invariant functions.

Statement recorded in the supplied source.

Is there a single event $\Omega_0$ of probability one such that for every $\omega\in\Omega_0$, every commuting system $(X,\mathcal X,\mu,T,S)$ and all bounded $f,g$, \[\lim_{N\to\infty}\frac1N\sum_{n=1}^NT^nf\,S^{a_n(\omega)}g=\mathbb E(f\mid\mathcal I_T)\mathbb E(g\mid\mathcal I_S)\quad\text{in }L^2(\mu)?\] Moreover, when $\sigma_j=j^{-\alpha}$ for any $\alpha\in(0,1)$, does a universal probability-one event give the same convergence for $\mu$-almost every point of every such system? [S2] The first question is reproduced with the stated nonincreasing scope. That literal formulation has a boundary obstruction: if $\sigma_j=1$ for every $j$, then $a_n=n$. On the probability circle $\mathbb R/\mathbb Z$ with Lebesgue measure, let $T=S$ be an irrational rotation, take $f(x)=e^{2\pi i x}$ and $g=\bar f$. This is an ergodic system, so both conditional expectations on the right vanish, whereas $T^nf\,S^ng=1$ for every $n$. Thus this literal first formulation has a negative answer; the separate power-law question retains its stated scope.

Short English statement

Do nonincreasing random-selection probabilities with $n\sigma_n\to\infty$ give the universal bilinear mean limit, and do all power-law selections also give the pointwise limit?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 4800032 (AMR-047-0032). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 N. Frantzikinakis, Some open problems on multiple ergodic averages (2016), Section 7 complete random-sequence setup, universal-almost-sure definition and full Problem 32, pp.34--35. https://arxiv.org/pdf/1103.3808v3.