Multiple ergodic averages — Problem 31

Definitions and mathematical statement

Let $(\sigma_k)_{k\ge1}$ be a nonincreasing sequence in $(0,1]$ with $k\sigma_k\to\infty$. On a probability space $(\Omega,\mathcal F,\mathbb P)$, take independent Bernoulli variables $\xi_k$ with $\mathbb P(\xi_k=1)=\sigma_k$. Almost surely the selected set is infinite; let $a_n(\omega)$ be its $n$th smallest member.

Prove that there is one event $\Omega_0$ of probability one such that for every $\omega\in\Omega_0$, every $\ell\ge1$, every probability-preserving commuting invertible system $(X,\mathcal B,\mu,T_1,\ldots,T_\ell)$, and all $f_i\in L^\infty(\mu)$, \[ \frac1N\sum_{n=1}^N\prod_{i=1}^\ell f_i\circ T_i^{a_n(\omega)} \quad\text{and}\quad \frac1N\sum_{n=1}^N\prod_{i=1}^\ell f_i\circ T_i^n \] have the same $L^2(\mu)$ limit. Also prove that each such selected sequence gives multiple recurrence: every $A$ with $\mu(A)>0$ admits some $n$ with $\mu(A\cap T_1^{a_n(\omega)}A\cap\cdots \cap T_\ell^{a_n(\omega)}A)>0$.

The full-probability event must be chosen before the dynamical system and observables are chosen; an event depending on each tested system would be weaker. Averages are normalized by the number of selected times $N$, not by the largest selected integer. No ergodicity or pointwise convergence is imposed. The sparsity condition and monotonicity concern the selection probabilities, not deterministic gaps between successive selected integers.

Short English statement

Does sufficiently frequent independent random sampling almost surely preserve every commuting multiple ergodic mean and recurrence property?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 4800031 (AMR-047-0031), AMR Open Problem Lists. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Frantzikinakis - Some open problems on multiple ergodic averages (2016), Problem 31. Original source indexed by AMR-047-0031. https://arxiv.org/abs/1103.3808.